Stony Brook University New Horizons in Twistor Theory ...people.maths.ox.ac.uk/lmason/New...
Transcript of Stony Brook University New Horizons in Twistor Theory ...people.maths.ox.ac.uk/lmason/New...
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Mass in
Kahler Geometry
Claude LeBrunStony Brook University
New Horizons in Twistor TheoryOxford, January 5, 2017
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Joint work with
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Joint work with
Hans-Joachim HeinUniversity of Maryland
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Joint work with
Hans-Joachim HeinFordham University
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Joint work with
Hans-Joachim HeinFordham University
Comm. Math. Phys. 347 (2016) 621–653.
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Definition. A complete, non-compact Rieman-nian n-manifold (Mn, g) is called asymptoticallyEuclidean (AE) if there is a compact set K ⊂Msuch that each component of M − K is diffeo-morphic to Rn −Dn
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Definition. A complete, non-compact Rieman-nian n-manifold (Mn, g) is called asymptoticallyEuclidean (AE) if there is a compact set K ⊂Msuch that each component of M − K is diffeo-morphic to Rn −Dn
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Definition. A complete, non-compact Rieman-nian n-manifold (Mn, g) is called asymptoticallyEuclidean (AE) if there is a compact set K ⊂Msuch that each component of M − K is diffeo-morphic to Rn −Dn in such a manner that
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Definition. A complete, non-compact Rieman-nian n-manifold (Mn, g) is called asymptoticallyEuclidean (AE) if there is a compact set K ⊂Msuch that each component of M − K is diffeo-morphic to Rn −Dn in such a manner that
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Definition. A complete, non-compact Rieman-nian n-manifold (Mn, g) is called asymptoticallyEuclidean (AE) if there is a compact set K ⊂Msuch that each component of M − K is diffeo-morphic to Rn −Dn in such a manner that
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Definition. A complete, non-compact Rieman-nian n-manifold (Mn, g) is called asymptoticallyEuclidean (AE) if there is a compact set K ⊂Msuch that each component of M − K is diffeo-morphic to Rn −Dn in such a manner that
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Definition. A complete, non-compact Rieman-nian n-manifold (Mn, g) is called asymptoticallyEuclidean (AE) if there is a compact set K ⊂Msuch that each component of M − K is diffeo-morphic to Rn −Dn in such a manner that
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~x
gjk = δjk + O(|x|1−n2−ε)
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Definition. A complete, non-compact Rieman-nian n-manifold (Mn, g) is called asymptoticallyEuclidean (AE) if there is a compact set K ⊂Msuch that each component of M − K is diffeo-morphic to Rn −Dn in such a manner that
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~x
gjk = δjk + O(|x|1−n2−ε)
gjk,` = O(|x|−n2−ε), s ∈ L1
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Definition. A complete, non-compact Rieman-nian n-manifold (Mn, g) is called asymptoticallyEuclidean (AE) if there is a compact set K ⊂Msuch that each component of M − K is diffeo-morphic to Rn −Dn in such a manner that
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~x
gjk = δjk + O(|x|1−n2−ε)
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Definition. A complete, non-compact Rieman-nian n-manifold (Mn, g) is called asymptoticallyEuclidean (AE) if there is a compact set K ⊂Msuch that each component of M − K is diffeo-morphic to Rn −Dn in such a manner that
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~x
gjk = δjk + O(|x|1−n2−ε)
gjk,` = O(|x|−n2−ε), scalar curvature ∈ L1
15
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Definition. A complete, non-compact Rieman-nian n-manifold (Mn, g) is called asymptoticallyEuclidean (AE) if there is a compact set K ⊂Msuch that each component of M − K is diffeo-morphic to Rn −Dn in such a manner that
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~x
gjk = δjk + O(|x|1−n2−ε)
gjk,` = O(|x|−n2−ε), s ∈ L1
16
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Definition. A complete, non-compact Rieman-nian n-manifold (Mn, g) is called asymptoticallyEuclidean (AE) if there is a compact set K ⊂Msuch that each component of M − K is diffeo-morphic to Rn −Dn in such a manner that
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~x
gjk = δjk + O(|x|1−n2−ε)
gjk,` = O(|x|−n2−ε), s ∈ L1
17
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Definition. A complete, non-compact Rieman-nian n-manifold (Mn, g) is called asymptoticallyEuclidean (AE) if there is a compact set K ⊂Msuch that each component of M − K is diffeo-morphic to Rn −Dn in such a manner that
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gjk = δjk + O(|x|1−n2−ε)
gjk,` = O(|x|−n2−ε), s ∈ L1
18
![Page 19: Stony Brook University New Horizons in Twistor Theory ...people.maths.ox.ac.uk/lmason/New Horizons/Claude-Lebrun.pdf · K ahler Geometry Claude LeBrun Stony Brook University New Horizons](https://reader036.fdocuments.net/reader036/viewer/2022062602/5edc2785ad6a402d6666b2f7/html5/thumbnails/19.jpg)
Definition. A complete, non-compact Rieman-nian n-manifold (Mn, g) is called asymptoticallyEuclidean (AE) if there is a compact set K ⊂Msuch that each component of M − K is diffeo-morphic to Rn −Dn in such a manner that
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gjk = δjk + O(|x|1−n2−ε)
gjk,` = O(|x|−n2−ε), s ∈ L1
19
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Definition. A complete, non-compact Rieman-nian n-manifold (Mn, g) is called asymptoticallyEuclidean (AE) if there is a compact set K ⊂Msuch that each component of M − K is diffeo-morphic to Rn −Dn in such a manner that
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gjk = δjk + O(|x|1−n2−ε)
gjk,` = O(|x|−n2−ε), s ∈ L1
20
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Definition. A complete, non-compact Rieman-nian n-manifold (Mn, g) is called asymptoticallyEuclidean (AE) if there is a compact set K ⊂Msuch that each component of M − K is diffeo-morphic to Rn −Dn in such a manner that
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gjk = δjk + O(|x|1−n2−ε)
gjk,` = O(|x|−n2−ε), s ∈ L1
21
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Definition. A complete, non-compact Rieman-nian n-manifold (Mn, g) is called asymptoticallyEuclidean (AE) if there is a compact set K ⊂Msuch that each component of M − K is diffeo-morphic to Rn −Dn in such a manner that
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gjk = δjk + O(|x|1−n2−ε)
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22
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Definition. A complete, non-compact Rieman-nian n-manifold (Mn, g) is called asymptoticallyEuclidean (AE) if there is a compact set K ⊂Msuch that each “end” component of is diffeo-morphic to Rn −Dn in such a manner that
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gjk = δjk + O(|x|1−n2−ε)
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23
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Definition. Complete, non-compact n-manifold(Mn, g) is asymptotically locally Euclidean (ALE)if ∃ compact set K ⊂ M such that M − K ≈∐i(R
n −Dn)/Γi, where Γi ⊂ O(n), such that
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24
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Definition. Complete, non-compact n-manifold(Mn, g) is asymptotically locally Euclidean (ALE)if ∃ compact set K ⊂ M such that M − K ≈∐i(R
n −Dn)/Γi, where Γi ⊂ O(n), such that
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25
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Definition. Complete, non-compact n-manifold(Mn, g) is asymptotically locally Euclidean (ALE)if ∃ compact set K ⊂ M such that M − K ≈∐i(R
n −Dn)/Γi, where Γi ⊂ O(n), such that
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Definition. Complete, non-compact n-manifold(Mn, g) is asymptotically locally Euclidean (ALE)if ∃ compact set K ⊂ M such that M − K ≈∐i(Rn −Dn)/Γi, where Γi ⊂ O(n), such that
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Definition. Complete, non-compact n-manifold(Mn, g) is asymptotically locally Euclidean (ALE)if ∃ compact set K ⊂ M such that M − K ≈∐i(Rn −Dn)/Γi, where Γi ⊂ O(n), such that
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Definition. Complete, non-compact n-manifold(Mn, g) is asymptotically locally Euclidean (ALE)if ∃ compact set K ⊂ M such that M − K ≈∐i(Rn −Dn)/Γi, where Γi ⊂ O(n), such that
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Definition. Complete, non-compact n-manifold(Mn, g) is asymptotically locally Euclidean (ALE)if ∃ compact set K ⊂ M such that M − K ≈∐i(Rn −Dn)/Γi, where Γi ⊂ O(n), such that
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30
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Definition. Complete, non-compact n-manifold(Mn, g) is asymptotically locally Euclidean (ALE)if ∃ compact set K ⊂ M such that M − K ≈∐i(Rn −Dn)/Γi, where Γi ⊂ O(n), such that
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gjk = δjk + O(|x|1−n2−ε)
gjk,` = O(|x|−n2−ε), s ∈ L1
31
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Why consider ALE spaces?
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Key examples:
33
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Key examples:
Term ALE coined by Gibbons & Hawking, 1979.
34
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Key examples:
Term ALE coined by Gibbons & Hawking, 1979.
They wrote down various explicit Ricci-flat ALE4-manifolds they called gravitational instantons.
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Key examples:
Term ALE coined by Gibbons & Hawking, 1979.
They wrote down various explicit Ricci-flat ALE4-manifolds they called gravitational instantons.
By contrast, any Ricci-flat AE manifold must beflat, by the Bishop-Gromov inequality. . .
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Key examples:
Term ALE coined by Gibbons & Hawking, 1979.
They wrote down various explicit Ricci-flat ALE4-manifolds they called gravitational instantons.
Their examples have just one end, with
37
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Key examples:
Term ALE coined by Gibbons & Hawking, 1979.
They wrote down various explicit Ricci-flat ALE4-manifolds they called gravitational instantons.
Their examples have just one end, with
Γ ∼= Z` ⊂ SU(2) ⊂ O(4).
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vv
vvv
Data: ` points in R3. =⇒ V with ∆V = 0
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vv
vvv
Data: ` points in R3. =⇒ V with ∆V = 0
V =∑j=1
1
2%j
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vv
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Data: ` points in R3. =⇒ V with ∆V = 0
V =∑j=1
1
2%j
F = ?dV curvature θ on P → R3 − pts.
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vv
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Data: ` points in R3. =⇒ V with ∆V = 0
g = V h + V −1θ2
F = ?dV curvature θ on P → R3 − pts.
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vv
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Data: ` points in R3. =⇒ V with ∆V = 0
g = V h + V −1θ2
on P . Then take M4 = Riemannian completion.
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Key examples:
Term ALE coined by Gibbons & Hawking, 1979.
They wrote down various explicit Ricci-flat ALE4-manifolds they called gravitational instantons.
Their examples have just one end, with
Γ ∼= Z` ⊂ SU(2) ⊂ O(4).
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Key examples:
Term ALE coined by Gibbons & Hawking, 1979.
They wrote down various explicit Ricci-flat ALE4-manifolds they called gravitational instantons.
Their examples have just one end, with
Γ ∼= Z` ⊂ SU(2) ⊂ O(4).
The G-H metrics are hyper-Kahler, and were soonrediscovered independently by Hitchin.
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Kahler metrics:
(Mn, g): Kahler ⇐⇒ holonomy ⊂ U(m)
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Kahler metrics:
(Mn, g): Kahler ⇐⇒ holonomy ⊂ U(m)
s
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Kahler metrics:
(Mn, g): Kahler ⇐⇒ holonomy ⊂ U(m)
s
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Kahler metrics:
(Mn, g): Kahler ⇐⇒ holonomy ⊂ U(m)
s
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Kahler metrics:
(Mn, g): Kahler ⇐⇒ holonomy ⊂ U(m)
s
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Kahler metrics:
(Mn, g): Kahler ⇐⇒ holonomy ⊂ U(m)
s
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Kahler metrics:
(Mn, g): Kahler ⇐⇒ holonomy ⊂ U(m)
s
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52
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Kahler metrics:
(Mn, g): Kahler ⇐⇒ holonomy ⊂ U(m)
s
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Kahler metrics:
(Mn, g): Kahler ⇐⇒ holonomy ⊂ U(m)
s
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Kahler metrics:
(Mn, g): Kahler ⇐⇒ holonomy ⊂ U(m)
s
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55
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Kahler metrics:
(Mn, g): Kahler ⇐⇒ holonomy ⊂ U(m)
s
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Kahler metrics:
(Mn, g): Kahler ⇐⇒ holonomy ⊂ U(m)
s
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Kahler metrics:
(Mn, g): Kahler ⇐⇒ holonomy ⊂ U(m)
s
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Kahler metrics:
(Mn, g): Kahler ⇐⇒ holonomy ⊂ U(m)
s
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Kahler metrics:
(Mn, g): Kahler ⇐⇒ holonomy ⊂ U(m)
s
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Kahler metrics:
(Mn, g): Kahler ⇐⇒ holonomy ⊂ U(m)
s
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Hyper-Kahler metrics:
(M4, g) hyper-Kahler ⇐⇒ holonomy ⊂ Sp(1)
s
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Hyper-Kahler metrics:
(M4, g) hyper-Kahler ⇐⇒ holonomy ⊂ Sp(1)
s
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Sp(1) = SU(2)
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Hyper-Kahler metrics:
(M4, g) hyper-Kahler ⇐⇒ holonomy ⊂ Sp(1)
s
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Sp(1) = SU(2)
⇐⇒ Λ+ flat and trivial.
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Hyper-Kahler metrics:
(M4, g) hyper-Kahler ⇐⇒ holonomy ⊂ Sp(1)
s
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Sp(1) = SU(2)
⇐⇒ Λ+ flat and trivial.
Locally, ⇐⇒ s = 0, r = 0, W+ = 0.
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Hyper-Kahler metrics:
(M4, g) hyper-Kahler ⇐⇒ holonomy ⊂ Sp(1)
s
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Sp(1) = SU(2)
Ricci-flat and Kahler,
for many different complex structures!
66
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Hyper-Kahler metrics:
(M4, g) hyper-Kahler ⇐⇒ holonomy ⊂ U(2)
s
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Sp(1) ⊂ U(2)
Ricci-flat and Kahler,
for many different complex structures!
67
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Hyper-Kahler metrics:
(M4, g) hyper-Kahler ⇐⇒ holonomy ⊂ Sp(1)
s
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Sp(1) = SU(2)
Ricci-flat and Kahler,
for many different complex structures!
68
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All these complex structures can be repackaged as
Penrose Twistor Space (Z6, J),
which is a complex 3-manifold
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All these complex structures can be repackaged as
Penrose Twistor Space (Z, J),
which is a complex 3-manifold
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70
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All these complex structures can be repackaged as
Penrose Twistor Space (Z, J),
which is a complex 3-manifold.
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71
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All these complex structures can be repackaged as
Penrose Twistor Space (Z, J),
which is a complex 3-manifold.
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Riemannian non-linear graviton construction.
72
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Hitchin’s Twistor Spaces:
73
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Hitchin’s Twistor Spaces:
H0(CP1,O(2)) = C3⊃ R3.
74
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Hitchin’s Twistor Spaces:
H0(CP1,O(2)) = C3 ⊃ R3.
75
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Hitchin’s Twistor Spaces:
H0(CP1,O(2)) = C3 ⊃ R3.
vv
vvv
So ` points determine P1, . . . , P` ∈ H0(CP1,O(2)).
76
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Hitchin’s Twistor Spaces:
H0(CP1,O(2)) = C3 ⊃ R3.
vv
vvv
So ` points determine P1, . . . , P` ∈ H0(CP1,O(2)).
Small resolution Z of Z ⊂ O(`)⊕O(`)⊕O(2)
77
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Hitchin’s Twistor Spaces:
H0(CP1,O(2)) = C3 ⊃ R3.
vv
vvv
So ` points determine P1, . . . , P` ∈ H0(CP1,O(2)).
Small resolution Z of Z ⊂ O(`)⊕O(`)⊕O(2)
xy = (z − P1) · · · (z − P`)
78
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Hitchin’s Twistor Spaces:
H0(CP1,O(2)) = C3 ⊃ R3.
vv
vvv
So ` points determine P1, . . . , P` ∈ H0(CP1,O(2)).
Small resolution Z of Z ⊂ O(`)⊕O(`)⊕O(2)
xy = (z − P1) · · · (z − P`)
79
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Hitchin’s Twistor Spaces:
H0(CP1,O(2)) = C3 ⊃ R3.
vv
vvv
So ` points determine P1, . . . , P` ∈ H0(CP1,O(2)).
Small resolution Z of Z ⊂ O(`)⊕O(`)⊕O(2)
xy = (z − P1) · · · (z − P`)is the twistor space of a Gibbons-Hawking metric.
80
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Key examples:
Term ALE coined by Gibbons & Hawking, 1979.
They wrote down various explicit Ricci-flat ALE4-manifolds they called gravitational instantons.
Their examples have just one end, with
Γ ∼= Z` ⊂ SU(2) ⊂ O(4).
The G-H metrics are hyper-Kahler, and were soonrediscovered independently by Hitchin.
81
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Key examples:
Term ALE coined by Gibbons & Hawking, 1979.
They wrote down various explicit Ricci-flat ALE4-manifolds they called gravitational instantons.
Their examples have just one end, with
Γ ∼= Z` ⊂ SU(2) ⊂ O(4).
The G-H metrics are hyper-Kahler, and were soonrediscovered independently by Hitchin.
Hitchin conjectured that similar metrics would existfor each finite Γ ⊂ SU(2).
82
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Key examples:
Term ALE coined by Gibbons & Hawking, 1979.
They wrote down various explicit Ricci-flat ALE4-manifolds they called gravitational instantons.
Their examples have just one end, with
Γ ∼= Z` ⊂ SU(2) ⊂ O(4).
The G-H metrics are hyper-Kahler, and were soonrediscovered independently by Hitchin.
Hitchin conjectured that similar metrics would existfor each finite Γ ⊂ SU(2).
This conjecture was proved by Kronheimer, 1986.
83
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Felix Klein, 1884: C2/Γ → C3
Zk+1 ←→ xy + zk+1 = 0
Dih∗k−2 ←→ x2 + z(y2 + zk−2) = 0
T ∗ ←→ x2 + y3 + z4 = 0
O∗ ←→ x2 + y3 + yz3 = 0
I∗ ←→ x2 + y3 + z5 = 0
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Zk+1←→Ak • • • •.......................................................................................................................................................
Dih∗k−2←→Dk • • • ••.................................................................................................................
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T ∗←→E6 • • • •••................................................................................................................................................................................................................
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O∗←→E7 • • • ••• •..................................................................................................................................................................................................................................................................
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I∗←→E8 • • • ••• • •....................................................................................................................................................................................................................................................................................................................
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Hyper-Kahler metrics:
(M4, g) hyper-Kahler ⇐⇒ holonomy ⊂ U(2)
s
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Hyper-Kahler metrics:
(M4, g) hyper-Kahler ⇐⇒ holonomy ⊂ U(2)
s
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In real dimension 4, any Kahler manifold satisfies
|W+|2 =s2
24
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Hyper-Kahler metrics:
(M4, g) hyper-Kahler ⇐⇒ holonomy ⊂ U(2)
s
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In real dimension 4, any Kahler manifold satisfies
|W+|2 =s2
24
so that W+ = 0 ⇐⇒ s = 0.
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Any scalar-flat Kahler surface (M4, g, J) has a
Penrose Twistor Space (Z, J),
which is once again a complex 3-manifold.
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Any scalar-flat Kahler surface (M4, g, J) has a
Penrose Twistor Space (Z, J),
which is once again a complex 3-manifold.
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Kahler case: holomorphic section of KZ−1/2
vanishing at D := (M,J) ∩ (M,−J).
90
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Any scalar-flat Kahler surface (M4, g, J) has a
Penrose Twistor Space (Z, J),
which is once again a complex 3-manifold.
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Kahler case: holomorphic section of KZ−1/2
vanishing at D := (M,J) ∩ (M,−J).
=⇒ Normal bundle of (M,J) in Z is KM−1.
91
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Any scalar-flat Kahler surface (M4, g, J) has a
Penrose Twistor Space (Z, J),
which is once again a complex 3-manifold.
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0→ T 1,0M → T 1,0Z|M → KM−1→ 0
=⇒ Normal bundle of (M,J) in Z is KM−1.
92
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Any scalar-flat Kahler surface (M4, g, J) has a
Penrose Twistor Space (Z, J),
which is once again a complex 3-manifold.
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0→ T 1,0M → T 1,0Z|M → KM−1→ 0
Extension class: [ω] ∈ H1(M,O(KM⊗T 1,0M)).
93
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Any scalar-flat Kahler surface (M4, g, J) has a
Penrose Twistor Space (Z, J),
which is once again a complex 3-manifold.
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0→ T 1,0M → T 1,0Z|M → KM−1→ 0
Extension class: [ω] ∈ H1(M,Ω1).
94
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Any scalar-flat Kahler surface (M4, g, J) has a
Penrose Twistor Space (Z, J),
which is once again a complex 3-manifold.
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0→ T 1,0M → T 1,0Z|M → KM−1→ 0
Extension class: [ω] ∈ H1(M,Ω1).
95
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Any scalar-flat Kahler surface (M4, g, J) has a
Penrose Twistor Space (Z, J),
which is once again a complex 3-manifold.
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0→ T 1,0M → T 1,0Z|M → KM−1→ 0
Kahler class: [ω] ∈ H1(M,Ω1).
96
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Any scalar-flat Kahler surface (M4, g, J) has a
Penrose Twistor Space (Z, J),
which is once again a complex 3-manifold.
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0→ T 1,0M → T 1,0Z|M → KM−1→ 0
Kahler form: ω = g(J ·, ·).
97
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Some AE/ALE Scalar-Flat Kahler Surfaces:
98
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Some AE/ALE Scalar-Flat Kahler Surfaces:
(L ’91)
99
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Some AE/ALE Scalar-Flat Kahler Surfaces:
100
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Some AE/ALE Scalar-Flat Kahler Surfaces:
f
vv
vvv
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Data: k + 1 points in H3. =⇒ V with ∆V = 0
101
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Some AE/ALE Scalar-Flat Kahler Surfaces:
f
vv
vvv
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Data: k + 1 points in H3. =⇒ V with ∆V = 0
V = 1 +`
e2%0 − 1+
k∑j=1
1
e2%j − 1
102
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Some AE/ALE Scalar-Flat Kahler Surfaces:
f
vv
vvv
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Data: k + 1 points in H3. =⇒ V with ∆V = 0
F = ?dV curvature θ on P → H3 − pts.
103
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Some AE/ALE Scalar-Flat Kahler Surfaces:
f
vv
vvv
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Data: k + 1 points in H3. =⇒ V with ∆V = 0
g =1
4 sinh2 %0
(V h + V −1θ2
)
104
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Some AE/ALE Scalar-Flat Kahler Surfaces:
f
vv
vvv
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Riemannian completion is ALE scalar-flat Kahler.
g =1
4 sinh2 %0
(V h + V −1θ2
)
105
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Some AE/ALE Scalar-Flat Kahler Surfaces:
f
vv
vvv
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Riemannian completion is AE ⇐⇒ ` = 1:
V = 1 +`
e2%0 − 1+
k∑j=1
1
e2%j − 1
106
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Twistor Spaces for These Metrics:
H0(CP1 × CP1,O(1, 1)) = C4⊃ R1,3 ⊃ H3
107
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Twistor Spaces for These Metrics:
H0(CP1 × CP1,O(1, 1)) = C4 ⊃ R1,3⊃ H3
108
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Twistor Spaces for These Metrics:
H0(CP1 × CP1,O(1, 1)) = C4 ⊃ R1,3 ⊃ H3
109
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Twistor Spaces for These Metrics:
H0(CP1 × CP1,O(1, 1)) = C4 ⊃ R1,3 ⊃ H3
f
vv
vvv
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So k + 1 points in H3 give rise to
110
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Twistor Spaces for These Metrics:
H0(CP1 × CP1,O(1, 1)) = C4 ⊃ R1,3 ⊃ H3
f
vv
vvv
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So k + 1 points in H3 give rise to
P0, P1, . . . , Pk ∈ H0(CP1 × CP1,O(1, 1)).
111
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Twistor Spaces for These Metrics:
H0(CP1 × CP1,O(1, 1)) = C4 ⊃ R1,3 ⊃ H3
112
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Twistor Spaces for These Metrics:
H0(CP1 × CP1,O(1, 1)) = C4 ⊃ R1,3 ⊃ H3
InO(k + `− 1, 1)⊕O(1, k + `− 1)→ CP1×CP1,
113
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Twistor Spaces for These Metrics:
H0(CP1 × CP1,O(1, 1)) = C4 ⊃ R1,3 ⊃ H3
InO(k + `− 1, 1)⊕O(1, k + `− 1)→ CP1×CP1,
let Z be the hypersurface
xy = P `0 P1 · · · Pk.
114
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Twistor Spaces for These Metrics:
H0(CP1 × CP1,O(1, 1)) = C4 ⊃ R1,3 ⊃ H3
InO(k + `− 1, 1)⊕O(1, k + `− 1)→ CP1×CP1,
let Z be the hypersurface
xy = P `0 P1 · · · Pk.
Then twistor space Z obtained from Z by
115
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Twistor Spaces for These Metrics:
H0(CP1 × CP1,O(1, 1)) = C4 ⊃ R1,3 ⊃ H3
InO(k + `− 1, 1)⊕O(1, k + `− 1)→ CP1×CP1,
let Z be the hypersurface
xy = P `0 P1 · · · Pk.
Then twistor space Z obtained from Z by
• removing curve in zero section cut out by P0,
116
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Twistor Spaces for These Metrics:
H0(CP1 × CP1,O(1, 1)) = C4 ⊃ R1,3 ⊃ H3
InO(k + `− 1, 1)⊕O(1, k + `− 1)→ CP1×CP1,
let Z be the hypersurface
xy = P `0 P1 · · · Pk.
Then twistor space Z obtained from Z by
• removing curve in zero section cut out by P0,
• adding two rational curves at infinity, and
117
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Twistor Spaces for These Metrics:
H0(CP1 × CP1,O(1, 1)) = C4 ⊃ R1,3 ⊃ H3
InO(k + `− 1, 1)⊕O(1, k + `− 1)→ CP1×CP1,
let Z be the hypersurface
xy = P `0 P1 · · · Pk.
Then twistor space Z obtained from Z by
• removing curve in zero section cut out by P0,
• adding two rational curves at infinity, and
•making small resolutions of isolated singularities.
118
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Any scalar-flat Kahler surface (M4, g, J) has a
Penrose Twistor Space (Z, J),
which is once again a complex 3-manifold.
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119
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Any scalar-flat Kahler surface (M4, g, J) has a
Penrose Twistor Space (Z, J),
which is once again a complex 3-manifold.
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Lots more ALE scalar-flat Kahler surfaces now known:
120
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Any scalar-flat Kahler surface (M4, g, J) has a
Penrose Twistor Space (Z, J),
which is once again a complex 3-manifold.
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rr Z
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Lots more ALE scalar-flat Kahler surfaces now known:
Joyce, Calderbank-Singer, Lock-Viaclovsky. . .
121
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Any scalar-flat Kahler surface (M4, g, J) has a
Penrose Twistor Space (Z, J),
which is once again a complex 3-manifold.
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r
rr Z
M 4
↓
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Lots more ALE scalar-flat Kahler surfaces now known:
Joyce, Calderbank-Singer, Lock-Viaclovsky. . .
But full classification remains an open problem.
122
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Definition. Complete, non-compact n-manifold(Mn, g) is asymptotically locally Euclidean (ALE)if ∃ compact set K ⊂ M such that M − K ≈∐i(Rn −Dn)/Γi, where Γi ⊂ O(n), such that
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....
gjk = δjk + O(|x|1−n2−ε)
gjk,` = O(|x|−n2−ε), s ∈ L1
123
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Definition. The mass (at a given end) of anALE n-manifold is defined to be
m(M, g) := lim%→∞
Γ(n2)
4(n− 1)πn/2
∫Σ(%)
[gij,i − gii,j
]νjαE
124
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Definition. The mass (at a given end) of anALE n-manifold is defined to be
m(M, g) := lim%→∞
Γ(n2)
4(n− 1)πn/2
∫Σ(%)
[gij,i − gii,j
]νjαE
125
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Definition. The mass (at a given end) of anALE n-manifold is defined to be
m(M, g) := lim%→∞
Γ(n2)
4(n− 1)πn/2
∫Σ(%)
[gij,i − gii,j
]νjαE
126
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Definition. The mass (at a given end) of anALE n-manifold is defined to be
m(M, g) := lim%→∞
Γ(n2)
4(n− 1)πn/2
∫Σ(%)
[gij,i − gii,j
]νjαE
127
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Definition. The mass (at a given end) of anALE n-manifold is defined to be
m(M, g) := lim%→∞
Γ(n2)
4(n− 1)πn/2
∫Σ(%)
[gij,i − gii,j
]νjαE
128
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Definition. The mass (at a given end) of anALE n-manifold is defined to be
m(M, g) := lim%→∞
Γ(n2)
4(n− 1)πn/2
∫Σ(%)
[gij,i − gii,j
]νjαE
where
129
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Definition. The mass (at a given end) of anALE n-manifold is defined to be
m(M, g) := lim%→∞
Γ(n2)
4(n− 1)πn/2
∫Σ(%)
[gij,i − gii,j
]νjαE
where
• Σ(%) ≈ Sn−1/Γi is given by |~x| = %;
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130
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Definition. The mass (at a given end) of anALE n-manifold is defined to be
m(M, g) := lim%→∞
Γ(n2)
4(n− 1)πn/2
∫Σ(%)
[gij,i − gii,j
]νjαE
where
• Σ(%) ≈ Sn−1/Γi is given by |~x| = %;
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131
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Definition. The mass (at a given end) of anALE n-manifold is defined to be
m(M, g) := lim%→∞
Γ(n2)
4(n− 1)πn/2
∫Σ(%)
[gij,i − gii,j
]νjαE
where
• Σ(%) ≈ Sn−1/Γi is given by |~x| = %;
• ν is the outpointing Euclidean unit normal;and
132
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Definition. The mass (at a given end) of anALE n-manifold is defined to be
m(M, g) := lim%→∞
Γ(n2)
4(n− 1)πn/2
∫Σ(%)
[gij,i − gii,j
]νjαE
where
• Σ(%) ≈ Sn−1/Γi is given by |~x| = %;
• ν is the outpointing Euclidean unit normal;and
• αE is the volume (n− 1)-form induced by theEuclidean metric.
133
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Definition. The mass (at a given end) of anALE n-manifold is defined to be
m(M, g) := lim%→∞
Γ(n2)
4(n− 1)πn/2
∫Σ(%)
[gij,i − gii,j
]νjαE
where
• Σ(%) ≈ Sn−1/Γi is given by |~x| = %;
• ν is the outpointing Euclidean unit normal;and
• αE is the volume (n− 1)-form induced by theEuclidean metric.
Seems to depend on choice of coordinates!
134
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Definition. The mass (at a given end) of anALE n-manifold is defined to be
m(M, g) := lim%→∞
Γ(n2)
4(n− 1)πn/2
∫Σ(%)
[gij,i − gii,j
]νjαE
where
• Σ(%) ≈ Sn−1/Γi is given by |~x| = %;
• ν is the outpointing Euclidean unit normal;and
• αE is the volume (n− 1)-form induced by theEuclidean metric.
135
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Definition. The mass (at a given end) of anALE n-manifold is defined to be
m(M, g) := lim%→∞
Γ(n2)
4(n− 1)πn/2
∫Σ(%)
[gij,i − gii,j
]νjαE
where
• Σ(%) ≈ Sn−1/Γi is given by |~x| = %;
• ν is the outpointing Euclidean unit normal;and
• αE is the volume (n− 1)-form induced by theEuclidean metric.
Bartnik/Chrusciel (1986): With weak fall-offconditions, the mass is well-defined & coordinateindependent.
136
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Definition. The mass (at a given end) of anALE n-manifold is defined to be
m(M, g) := lim%→∞
Γ(n2)
4(n− 1)πn/2
∫Σ(%)
[gij,i − gii,j
]νjαE
where
• Σ(%) ≈ Sn−1/Γi is given by |~x| = %;
• ν is the outpointing Euclidean unit normal;and
• αE is the volume (n− 1)-form induced by theEuclidean metric.
Bartnik/Chrusciel (1986): With weak fall-offconditions, the mass is well-defined & coordinateindependent.
137
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Definition. The mass (at a given end) of anALE n-manifold is defined to be
m(M, g) := lim%→∞
Γ(n2)
4(n− 1)πn/2
∫Σ(%)
[gij,i − gii,j
]νjαE
where
• Σ(%) ≈ Sn−1/Γi is given by |~x| = %;
• ν is the outpointing Euclidean unit normal;and
• αE is the volume (n− 1)-form induced by theEuclidean metric.
gjk = δjk + O(|x|1−n2−ε)
gjk,` = O(|x|−n2−ε), s ∈ L1
138
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Definition. The mass (at a given end) of anALE n-manifold is defined to be
m(M, g) := lim%→∞
Γ(n2)
4(n− 1)πn/2
∫Σ(%)
[gij,i − gii,j
]νjαE
where
• Σ(%) ≈ Sn−1/Γi is given by |~x| = %;
• ν is the outpointing Euclidean unit normal;and
• αE is the volume (n− 1)-form induced by theEuclidean metric.
Bartnik/Chrusciel (1986): With weak fall-offconditions, the mass is well-defined & coordinateindependent.
139
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Definition. The mass (at a given end) of anALE n-manifold is defined to be
m(M, g) := lim%→∞
Γ(n2)
4(n− 1)πn/2
∫Σ(%)
[gij,i − gii,j
]νjαE
where
• Σ(%) ≈ Sn−1/Γi is given by |~x| = %;
• ν is the outpointing Euclidean unit normal;and
• αE is the volume (n− 1)-form induced by theEuclidean metric.
Bartnik/Chrusciel (1986): With weak fall-offconditions, the mass is well-defined & coordinateindependent.
140
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Motivation:
141
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Motivation:
When n = 3, ADM mass in general relativity.
142
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Motivation:
When n = 3, ADM mass in general relativity.
Reads off “apparent mass” from strength of thegravitational field far from an isolated source.
143
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Motivation:
When n = 3, ADM mass in general relativity.
Reads off “apparent mass” from strength of thegravitational field far from an isolated source.
144
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Motivation:
When n = 3, ADM mass in general relativity.
Reads off “apparent mass” from strength of thegravitational field far from an isolated source.
145
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Motivation:
When n = 3, ADM mass in general relativity.
Reads off “apparent mass” from strength of thegravitational field far from an isolated source.
In any dimension, reproduces “mass” of t = 0hypersurface in (n + 1)-dimensional Schwarzschild
146
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Motivation:
When n = 3, ADM mass in general relativity.
Reads off “apparent mass” from strength of thegravitational field far from an isolated source.
In any dimension, reproduces “mass” of t = 0hypersurface in (n + 1)-dimensional Schwarzschild
g = −(
1− 2m%n−2
)dt2+
(1− 2m
%n−2
)−1
d%2+%2hSn−1
147
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Motivation:
When n = 3, ADM mass in general relativity.
Reads off “apparent mass” from strength of thegravitational field far from an isolated source.
In any dimension, reproduces “mass” of t = 0hypersurface in (n + 1)-dimensional Schwarzschild
g =
(1− 2m
%n−2
)−1
d%2+%2hSn−1
148
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Motivation:
When n = 3, ADM mass in general relativity.
Reads off “apparent mass” from strength of thegravitational field far from an isolated source.
In any dimension, reproduces “mass” of t = 0hypersurface in (n + 1)-dimensional Schwarzschild
g =
(1− 2m
%n−2
)−1
d%2+%2hSn−1
Two such regions fit together to formthe wormhole metric. Scalar-flat,AE, two ends. Not Ricci-flat, butconformally flat. Same mass m atboth ends: “size of throat.”
149
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Motivation:
When n = 3, ADM mass in general relativity.
Reads off “apparent mass” from strength of thegravitational field far from an isolated source.
In any dimension, reproduces “mass” of t = 0hypersurface in (n + 1)-dimensional Schwarzschild
g =
(1− 2m
%n−2
)−1
d%2+%2hSn−1
Two such regions fit together to formthe wormhole metric. Scalar-flat,AE, two ends. Not Ricci-flat, butconformally flat. Same mass m atboth ends: “size of throat.”
150
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Motivation:
When n = 3, ADM mass in general relativity.
Reads off “apparent mass” from strength of thegravitational field far from an isolated source.
In any dimension, reproduces “mass” of t = 0hypersurface in (n + 1)-dimensional Schwarzschild
g =
(1− 2m
%n−2
)−1
d%2+%2hSn−1
Two such regions fit together to formthe wormhole metric. Scalar-flat,AE, two ends. Not Ricci-flat, butconformally flat. Same mass m atboth ends: “size of throat.”
151
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Motivation:
When n = 3, ADM mass in general relativity.
Reads off “apparent mass” from strength of thegravitational field far from an isolated source.
In any dimension, reproduces “mass” of t = 0hypersurface in (n + 1)-dimensional Schwarzschild
g =
(1− 2m
%n−2
)−1
d%2+%2hSn−1
Two such regions fit together to formthe wormhole metric. Scalar-flat,AE, two ends. Not Ricci-flat, butconformally flat. Same mass m atboth ends: “size of throat.”
152
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Motivation:
When n = 3, ADM mass in general relativity.
Reads off “apparent mass” from strength of thegravitational field far from an isolated source.
In any dimension, reproduces “mass” of t = 0hypersurface in (n + 1)-dimensional Schwarzschild
g =
(1− 2m
%n−2
)−1
d%2+%2hSn−1
Two such regions fit together to formthe wormhole metric. Scalar-flat,AE, two ends. Not Ricci-flat, butconformally flat. Same mass m atboth ends: “size of throat.”
153
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Motivation:
When n = 3, ADM mass in general relativity.
Reads off “apparent mass” from strength of thegravitational field far from an isolated source.
In any dimension, reproduces “mass” of t = 0hypersurface in (n + 1)-dimensional Schwarzschild
g =
(1− 2m
%n−2
)−1
d%2+%2hSn−1
154
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Motivation:
When n = 3, ADM mass in general relativity.
Reads off “apparent mass” from strength of thegravitational field far from an isolated source.
In any dimension, reproduces “mass” of t = 0hypersurface in (n + 1)-dimensional Schwarzschild
g =
(1− 2m
%n−2
)−1
d%2+%2hSn−1
But any Ricci-flat ALE manifold has mass zero.
155
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Motivation:
When n = 3, ADM mass in general relativity.
Reads off “apparent mass” from strength of thegravitational field far from an isolated source.
In any dimension, reproduces “mass” of t = 0hypersurface in (n + 1)-dimensional Schwarzschild
g =
(1− 2m
%n−2
)−1
d%2+%2hSn−1
But any Ricci-flat ALE manifold has mass zero.
Bartnik: Ricci-flat =⇒ faster fall-off of metric!
156
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Motivation:
When n = 3, ADM mass in general relativity.
Reads off “apparent mass” from strength of thegravitational field far from an isolated source.
In any dimension, reproduces “mass” of t = 0hypersurface in (n + 1)-dimensional Schwarzschild
g =
(1− 2m
%n−2
)−1
d%2+%2hSn−1
But any Ricci-flat ALE manifold has mass zero.
Bartnik: Ricci-flat =⇒ faster fall-off of metric!
=⇒ “gravitational instantons” have mass zero.
157
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Motivation:
When n = 3, ADM mass in general relativity.
Reads off “apparent mass” from strength of thegravitational field far from an isolated source.
In any dimension, reproduces “mass” of t = 0hypersurface in (n + 1)-dimensional Schwarzschild
g =
(1− 2m
%n−2
)−1
d%2+%2hSn−1
Scalar-flat-Kahler Burns metric on C2⊂ C2 × CP1
158
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Motivation:
When n = 3, ADM mass in general relativity.
Reads off “apparent mass” from strength of thegravitational field far from an isolated source.
In any dimension, reproduces “mass” of t = 0hypersurface in (n + 1)-dimensional Schwarzschild
g =
(1− 2m
%n−2
)−1
d%2+%2hSn−1
Scalar-flat-Kahler Burns metric on C2 ⊂ C2×CP1
159
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Motivation:
When n = 3, ADM mass in general relativity.
Reads off “apparent mass” from strength of thegravitational field far from an isolated source.
160
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Motivation:
When n = 3, ADM mass in general relativity.
Reads off “apparent mass” from strength of thegravitational field far from an isolated source.
In any dimension, reproduces “mass” of t = 0 hy-persurface in (n + 1)-dimensional Schwarzschild
g =
(1− 2m
%n−2
)−1
d%2+%2hSn−1
Scalar-flat-Kahler Burns metric on C2 ⊂ C2×CP1
161
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Motivation:
When n = 3, ADM mass in general relativity.
Reads off “apparent mass” from strength of thegravitational field far from an isolated source.
In any dimension, reproduces “mass” of t = 0 hy-persurface in (n + 1)-dimensional Schwarzschild
g =
(1− 2m
%n−2
)−1
d%2+%2hSn−1
Scalar-flat-Kahler Burns metric on C2 ⊂ C2×CP1
162
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Motivation:
When n = 3, ADM mass in general relativity.
Reads off “apparent mass” from strength of thegravitational field far from an isolated source.
In any dimension, reproduces “mass” of t = 0 hy-persurface in (n + 1)-dimensional Schwarzschild
g =
(1− 2m
%n−2
)−1
d%2+%2hSn−1
Scalar-flat-Kahler Burns metric on C2 ⊂ C2×CP1:
ω =i
2∂∂ [u + 3m log u] , u = |z1|2 + |z2|2
163
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Motivation:
When n = 3, ADM mass in general relativity.
Reads off “apparent mass” from strength of thegravitational field far from an isolated source.
In any dimension, reproduces “mass” of t = 0 hy-persurface in (n + 1)-dimensional Schwarzschild
g =
(1− 2m
%n−2
)−1
d%2+%2hSn−1
Scalar-flat-Kahler Burns metric on C2 ⊂ C2×CP1:
ω =i
2∂∂ [u + 3m log u] , u = |z1|2 + |z2|2
164
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Motivation:
When n = 3, ADM mass in general relativity.
Reads off “apparent mass” from strength of thegravitational field far from an isolated source.
In any dimension, reproduces “mass” of t = 0 hy-persurface in (n + 1)-dimensional Schwarzschild
g =
(1− 2m
%n−2
)−1
d%2+%2hSn−1
Scalar-flat-Kahler Burns metric on C2 ⊂ C2×CP1:
ω =i
2∂∂ [u + 3m log u] , u = |z1|2 + |z2|2
also has mass m .
165
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Positive Mass Conjecture:
166
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Positive Mass Conjecture:
Any AE manifold with s ≥ 0 has m ≥ 0.
167
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Positive Mass Conjecture:
Any AE manifold with s ≥ 0 has m ≥ 0.
168
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Positive Mass Conjecture:
Any AE manifold with s ≥ 0 has m ≥ 0.
169
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Positive Mass Conjecture:
Any AE manifold with s ≥ 0 has m ≥ 0.
Physical intuition:
170
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Positive Mass Conjecture:
Any AE manifold with s ≥ 0 has m ≥ 0.
Physical intuition:
Local matter density ≥ 0 =⇒ total mass ≥ 0.
171
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Positive Mass Conjecture:
Any AE manifold with s ≥ 0 has m ≥ 0.
Physical intuition:
Local matter density ≥ 0 =⇒ total mass ≥ 0.
172
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Positive Mass Conjecture:
Any AE manifold with s ≥ 0 has m ≥ 0.
173
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Positive Mass Conjecture:
Any AE manifold with s ≥ 0 has m ≥ 0.
Schoen-Yau 1979:
174
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Positive Mass Conjecture:
Any AE manifold with s ≥ 0 has m ≥ 0.
Schoen-Yau 1979:
Proved in dimension n ≤ 7.
175
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Positive Mass Conjecture:
Any AE manifold with s ≥ 0 has m ≥ 0.
Schoen-Yau 1979:
Proved in dimension n ≤ 7.
Witten 1981:
176
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Positive Mass Conjecture:
Any AE manifold with s ≥ 0 has m ≥ 0.
Schoen-Yau 1979:
Proved in dimension n ≤ 7.
Witten 1981:
Proved for spin manifolds (implicitly, for any n.)
177
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Positive Mass Conjecture:
Any AE manifold with s ≥ 0 has m ≥ 0.
Schoen-Yau 1979:
Proved in dimension n ≤ 7.
Witten 1981:
Proved for spin manifolds (implicitly, for any n).
178
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Positive Mass Conjecture:
Any AE manifold with s ≥ 0 has m ≥ 0.
Schoen-Yau 1979:
Proved in dimension n ≤ 7.
Witten 1981:
Proved for spin manifolds (implicitly, for any n).
Hawking-Pope 1978:
179
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Positive Mass Conjecture:
Any AE manifold with s ≥ 0 has m ≥ 0.
Schoen-Yau 1979:
Proved in dimension n ≤ 7.
Witten 1981:
Proved for spin manifolds (implicitly, for any n).
Hawking-Pope 1978:
Conjectured true in ALE case, too.
180
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Positive Mass Conjecture:
Any AE manifold with s ≥ 0 has m ≥ 0.
Schoen-Yau 1979:
Proved in dimension n ≤ 7.
Witten 1981:
Proved for spin manifolds (implicitly, for any n).
Hawking-Pope 1978:
Conjectured true in ALE case, too.
L 1987:
181
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Positive Mass Conjecture:
Any AE manifold with s ≥ 0 has m ≥ 0.
Schoen-Yau 1979:
Proved in dimension n ≤ 7.
Witten 1981:
Proved for spin manifolds (implicitly, for any n).
Hawking-Pope 1978:
Conjectured true in ALE case, too.
L 1987:
ALE counter-examples.
182
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Positive Mass Conjecture:
Any AE manifold with s ≥ 0 has m ≥ 0.
Schoen-Yau 1979:
Proved in dimension n ≤ 7.
Witten 1981:
Proved for spin manifolds (implicitly, for any n).
Hawking-Pope 1978:
Conjectured true in ALE case, too.
L 1987:
ALE counter-examples.
Scalar-flat Kahler metrics
183
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Positive Mass Conjecture:
Any AE manifold with s ≥ 0 has m ≥ 0.
Schoen-Yau 1979:
Proved in dimension n ≤ 7.
Witten 1981:
Proved for spin manifolds (implicitly, for any n).
Hawking-Pope 1978:
Conjectured true in ALE case, too.
L 1987:
ALE counter-examples.
Scalar-flat Kahler metrics
on line bundles L→ CP1 of Chern-class ≤ −3.
184
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Mass of ALE Kahler manifolds?
185
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Mass of ALE Kahler manifolds?
Scalar-flat Kahler case?
186
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Mass of ALE Kahler manifolds?
Scalar-flat Kahler case?
Lemma. Any ALE Kahler manifold has onlyone end.
187
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Mass of ALE Kahler manifolds?
Scalar-flat Kahler case?
Lemma. Any ALE Kahler manifold has onlyone end.
188
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Mass of ALE Kahler manifolds?
Scalar-flat Kahler case?
Lemma. Any ALE Kahler manifold has onlyone end.
189
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Mass of ALE Kahler manifolds?
Scalar-flat Kahler case?
Lemma. Any ALE Kahler manifold has onlyone end.
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190
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Mass of ALE Kahler manifolds?
Scalar-flat Kahler case?
Lemma. Any ALE Kahler manifold has onlyone end.
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n = 2m ≥ 4
191
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Mass of ALE Kahler manifolds?
Scalar-flat Kahler case?
Lemma. Any ALE Kahler manifold has onlyone end.
192
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Mass of ALE Kahler manifolds?
Scalar-flat Kahler case?
Lemma. Any ALE Kahler manifold has onlyone end.
Main Point:
193
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Mass of ALE Kahler manifolds?
Scalar-flat Kahler case?
Lemma. Any ALE Kahler manifold has onlyone end.
Main Point:
Mass of an ALE Kahler manifold is unambiguous.
194
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Mass of ALE Kahler manifolds?
Scalar-flat Kahler case?
Lemma. Any ALE Kahler manifold has onlyone end.
Main Point:
Mass of an ALE Kahler manifold is unambiguous.
Does not depend on the choice of an end!
195
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We begin with the scalar-flat Kahler case.
196
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We begin with the scalar-flat Kahler case.
Theorem A. The mass of an ALE scalar-flatKahler manifold is a topological invariant.
197
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We begin with the scalar-flat Kahler case.
Theorem A. The mass of an ALE scalar-flatKahler manifold is a topological invariant.
198
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We begin with the scalar-flat Kahler case.
Theorem A. The mass of an ALE scalar-flatKahler manifold is a topological invariant.
199
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We begin with the scalar-flat Kahler case.
Theorem A. The mass of an ALE scalar-flatKahler manifold is a topological invariant.
200
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We begin with the scalar-flat Kahler case.
Theorem A. The mass of an ALE scalar-flatKahler manifold is a topological invariant.
201
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We begin with the scalar-flat Kahler case.
Theorem A. The mass of an ALE scalar-flatKahler manifold is a topological invariant.
That is, m(M, g, J) is completely determined by
202
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We begin with the scalar-flat Kahler case.
Theorem A. The mass of an ALE scalar-flatKahler manifold is a topological invariant.
That is, m(M, g, J) is completely determined by
203
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We begin with the scalar-flat Kahler case.
Theorem A. The mass of an ALE scalar-flatKahler manifold is a topological invariant.
That is, m(M, g, J) is completely determined by
• the smooth manifold M ,
204
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We begin with the scalar-flat Kahler case.
Theorem A. The mass of an ALE scalar-flatKahler manifold is a topological invariant.
That is, m(M, g, J) is completely determined by
• the smooth manifold M ,
• the first Chern class c1 = c1(M,J) ∈ H2(M)of the complex structure, and
205
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We begin with the scalar-flat Kahler case.
Theorem A. The mass of an ALE scalar-flatKahler manifold is a topological invariant.
That is, m(M, g, J) is completely determined by
• the smooth manifold M ,
• the first Chern class c1 = c1(M,J) ∈ H2(M)of the complex structure, and
• the Kahler class [ω] ∈ H2(M) of the metric.
206
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We begin with the scalar-flat Kahler case.
Theorem A. The mass of an ALE scalar-flatKahler manifold is a topological invariant.
That is, m(M, g, J) is completely determined by
• the smooth manifold M ,
• the first Chern class c1 = c1(M,J) ∈ H2(M)of the complex structure, and
• the Kahler class [ω] ∈ H2(M) of the metric.
In fact, we will see that there is an explicit formulafor the mass in terms of these data!
207
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The explicit formula reproduces the mass in caseswhere it previously had been laboriously computedfrom the definition. But it also allows one to quicklyread it off quite generally.
208
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The explicit formula reproduces the mass in caseswhere it previously had been laboriously computedfrom the definition. But it also allows one to quicklyread it off quite generally.
209
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The explicit formula reproduces the mass in caseswhere it previously had been laboriously computedfrom the definition. But it also allows one to quicklyread it off quite generally.
Corollary, suggested by Cristiano Spotti:
210
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The explicit formula reproduces the mass in caseswhere it previously had been laboriously computedfrom the definition. But it also allows one to quicklyread it off quite generally.
Corollary, suggested by Cristiano Spotti:
Theorem B. Let (M4, g, J) be an ALE scalar-flat Kahler surface, and suppose that (M, g) isthe minimal resolution of a surface singularity.Then m(M, g) ≤ 0, with = iff g is Ricci-flat.
211
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The explicit formula reproduces the mass in caseswhere it previously had been laboriously computedfrom the definition. But it also allows one to quicklyread it off quite generally.
Corollary, suggested by Cristiano Spotti:
Theorem B. Let (M4, g, J) be an ALE scalar-flat Kahler surface, and suppose that (M, g) isthe minimal resolution of a surface singularity.Then m(M, g) ≤ 0, with = iff g is Ricci-flat.
212
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The explicit formula reproduces the mass in caseswhere it previously had been laboriously computedfrom the definition. But it also allows one to quicklyread it off quite generally.
Corollary, suggested by Cristiano Spotti:
Theorem B. Let (M4, g, J) be an ALE scalar-flat Kahler surface, and suppose that (M,J) isthe minimal resolution of a surface singularity.Then m(M, g) ≤ 0, with = iff g is Ricci-flat.
213
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The explicit formula reproduces the mass in caseswhere it previously had been laboriously computedfrom the definition. But it also allows one to quicklyread it off quite generally.
Corollary, suggested by Cristiano Spotti:
Theorem B. Let (M4, g, J) be an ALE scalar-flat Kahler surface, and suppose that (M,J) isthe minimal resolution of a surface singularity.Then m(M, g) ≤ 0, with = iff g is Ricci-flat.
214
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The explicit formula reproduces the mass in caseswhere it previously had been laboriously computedfrom the definition. But it also allows one to quicklyread it off quite generally.
Corollary, suggested by Cristiano Spotti:
Theorem B. Let (M4, g, J) be an ALE scalar-flat Kahler surface, and suppose that (M,J) isthe minimal resolution of a surface singularity.Then m(M, g) ≤ 0, with = iff g is Ricci-flat.
215
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The explicit formula reproduces the mass in caseswhere it previously had been laboriously computedfrom the definition. But it also allows one to quicklyread it off quite generally.
Corollary, suggested by Cristiano Spotti:
Theorem B. Let (M4, g, J) be an ALE scalar-flat Kahler surface, and suppose that (M,J) isthe minimal resolution of a surface singularity.Then m(M, g) ≤ 0, with = iff g is Ricci-flat.
Note that minimality is essential here.
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The explicit formula reproduces the mass in caseswhere it previously had been laboriously computedfrom the definition. But it also allows one to quicklyread it off quite generally.
Corollary, suggested by Cristiano Spotti:
Theorem B. Let (M4, g, J) be an ALE scalar-flat Kahler surface, and suppose that (M,J) isthe minimal resolution of a surface singularity.Then m(M, g) ≤ 0, with = iff g is Ricci-flat.
Note that minimality is essential here.
Non-minimal resolutions typically admit families ofsuch metrics for which the mass can be continuouslydeformed from negative to positive.
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Explicit formula depends on a topological fact:
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Explicit formula depends on a topological fact:
Lemma. Let (M, g) be any ALE manifold of realdimension n ≥ 4. x Then the natural map
219
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Explicit formula depends on a topological fact:
Lemma. Let (M, g) be any ALE manifold of realdimension n ≥ 4. x Then the natural map
220
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Explicit formula depends on a topological fact:
Lemma. Let (M, g) be any ALE manifold of realdimension n ≥ 4. Then the natural map
221
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Explicit formula depends on a topological fact:
Lemma. Let (M, g) be any ALE manifold of realdimension n ≥ 4. Then the natural map
222
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Explicit formula depends on a topological fact:
Lemma. Let (M, g) be any ALE manifold of realdimension n ≥ 4. Then the natural map
H2c (M)→ H2
dR(M)
is an isomorphism.
223
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Explicit formula depends on a topological fact:
Lemma. Let (M, g) be any ALE manifold of realdimension n ≥ 4. Then the natural map
H2c (M)→ H2
dR(M)
is an isomorphism.
224
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Explicit formula depends on a topological fact:
Lemma. Let (M, g) be any ALE manifold of realdimension n ≥ 4. Then the natural map
H2c (M)→ H2
dR(M)
is an isomorphism.
Here
Hpc (M) :=
ker d : Epc (M)→ Ep+1c (M)
dEp−1c (M)
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Explicit formula depends on a topological fact:
Lemma. Let (M, g) be any ALE manifold of realdimension n ≥ 4. Then the natural map
H2c (M)→ H2
dR(M)
is an isomorphism.
Here
Hpc (M) :=
ker d : Epc (M)→ Ep+1c (M)
dEp−1c (M)
where
Epc (M) := Smooth, compactly supported p-forms on M.
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Explicit formula depends on a topological fact:
Lemma. Let (M, g) be any ALE manifold of realdimension n ≥ 4. Then the natural map
H2c (M)→ H2
dR(M)
is an isomorphism.
227
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Explicit formula depends on a topological fact:
Lemma. Let (M, g) be any ALE manifold of realdimension n ≥ 4. Then the natural map
H2c (M)→ H2
dR(M)
is an isomorphism.
Definition. If (M, g, J) is any ALE Kahler man-ifold, we will use
228
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Explicit formula depends on a topological fact:
Lemma. Let (M, g) be any ALE manifold of realdimension n ≥ 4. Then the natural map
H2c (M)→ H2
dR(M)
is an isomorphism.
Definition. If (M, g, J) is any ALE Kahler man-ifold, we will use
229
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Explicit formula depends on a topological fact:
Lemma. Let (M, g) be any ALE manifold of realdimension n ≥ 4. Then the natural map
H2c (M)→ H2
dR(M)
is an isomorphism.
Definition. If (M, g, J) is any ALE Kahler man-ifold, we will use
230
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Explicit formula depends on a topological fact:
Lemma. Let (M, g) be any ALE manifold of realdimension n ≥ 4. Then the natural map
H2c (M)→ H2
dR(M)
is an isomorphism.
Definition. If (M, g, J) is any ALE Kahler man-ifold, we will use
♣ : H2dR(M)→ H2
c (M)
231
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Explicit formula depends on a topological fact:
Lemma. Let (M, g) be any ALE manifold of realdimension n ≥ 4. Then the natural map
H2c (M)→ H2
dR(M)
is an isomorphism.
Definition. If (M, g, J) is any ALE Kahler man-ifold, we will use
♣ : H2dR(M)→ H2
c (M)
to denote the inverse of the natural map
232
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Explicit formula depends on a topological fact:
Lemma. Let (M, g) be any ALE manifold of realdimension n ≥ 4. Then the natural map
H2c (M)→ H2
dR(M)
is an isomorphism.
Definition. If (M, g, J) is any ALE Kahler man-ifold, we will use
♣ : H2dR(M)→ H2
c (M)
to denote the inverse of the natural map
H2c (M)→ H2
dR(M)
induced by the inclusion of compactly supportedsmooth forms into all forms.
233
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Explicit formula depends on a topological fact:
Lemma. Let (M, g) be any ALE manifold of realdimension n ≥ 4. Then the natural map
H2c (M)→ H2
dR(M)
is an isomorphism.
Definition. If (M, g, J) is any ALE Kahler man-ifold, we will use
♣ : H2dR(M)→ H2
c (M)
to denote the inverse of the natural map
H2c (M)→ H2
dR(M)
induced by the inclusion of compactly supportedsmooth forms into all forms.
234
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We can now state our mass formula:
Theorem C. Any ALE Kahler manifold (M, g, J)of complex dimension m has mass given by
235
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We can now state our mass formula:
Theorem C. Any ALE Kahler manifold (M, g, J)ofcomplex dimension m has mass given by
236
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We can now state our mass formula:
Theorem C. Any ALE Kahler manifold (M, g, J)of complex dimension m has mass given by
237
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We can now state our mass formula:
Theorem C. Any ALE Kahler manifold (M, g, J)of complex dimension m has mass given by
238
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We can now state our mass formula:
Theorem C. Any ALE Kahler manifold (M, g, J)of complex dimension m has mass given by
m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1
+(m− 1)!
4(2m− 1)πm
∫Msgdµg
239
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We can now state our mass formula:
Theorem C. Any ALE Kahler manifold (M, g, J)of complex dimension m has mass given by
m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1
+(m− 1)!
4(2m− 1)πm
∫Msgdµg
240
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We can now state our mass formula:
Theorem C. Any ALE Kahler manifold (M, g, J)of complex dimension m has mass given by
m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1
+(m− 1)!
4(2m− 1)πm
∫Msgdµg
241
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We can now state our mass formula:
Theorem C. Any ALE Kahler manifold (M, g, J)of complex dimension m has mass given by
m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1
+(m− 1)!
4(2m− 1)πm
∫Msgdµg
242
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We can now state our mass formula:
Theorem C. Any ALE Kahler manifold (M, g, J)of complex dimension m has mass given by
m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1
+(m− 1)!
4(2m− 1)πm
∫Msgdµg
243
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We can now state our mass formula:
Theorem C. Any ALE Kahler manifold (M, g, J)of complex dimension m has mass given by
m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1
+(m− 1)!
4(2m− 1)πm
∫Msgdµg
where
244
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We can now state our mass formula:
Theorem C. Any ALE Kahler manifold (M, g, J)of complex dimension m has mass given by
m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1
+(m− 1)!
4(2m− 1)πm
∫Msgdµg
where
• s = scalar curvature;
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We can now state our mass formula:
Theorem C. Any ALE Kahler manifold (M, g, J)of complex dimension m has mass given by
m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1
+(m− 1)!
4(2m− 1)πm
∫Msgdµg
where
• s = scalar curvature;
• dµ = metric volume form;
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We can now state our mass formula:
Theorem C. Any ALE Kahler manifold (M, g, J)of complex dimension m has mass given by
m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1
+(m− 1)!
4(2m− 1)πm
∫Msgdµg
where
• s = scalar curvature;
• dµ = metric volume form;
• c1 = c1(M,J) ∈ H2(M) is first Chern class;
247
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We can now state our mass formula:
Theorem C. Any ALE Kahler manifold (M, g, J)of complex dimension m has mass given by
m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1
+(m− 1)!
4(2m− 1)πm
∫Msgdµg
where
• s = scalar curvature;
• dµ = metric volume form;
• c1 = c1(M,J) ∈ H2(M) is first Chern class;
• [ω] ∈ H2(M) is Kahler class of (g, J); and
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We can now state our mass formula:
Theorem C. Any ALE Kahler manifold (M, g, J)of complex dimension m has mass given by
m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1
+(m− 1)!
4(2m− 1)πm
∫Msgdµg
where
• s = scalar curvature;
• dµ = metric volume form;
• c1 = c1(M,J) ∈ H2(M) is first Chern class;
• [ω] ∈ H2(M) is Kahler class of (g, J); and
• 〈 , 〉 is pairing between H2c (M) and H2m−2(M).
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m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1
+(m− 1)!
4(2m− 1)πm
∫Msgdµg
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4πm(2m−1)(m−1)!
m(M, g) = − 4π(m−1)!
〈♣(c1), [ω]m−1〉+∫Msgdµg
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For a compact Kahler manifold (M2m, g, J),
∫Msgdµg =
4π
(m− 1)!〈c1, [ω]m−1〉
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For a compact Kahler manifold (M2m, g, J),
0 = − 4π
(m− 1)!〈c1, [ω]m−1〉 +
∫Msgdµg
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For an ALE Kahler manifold (M2m, g, J),
4πm(2m−1)(m−1)!
m(M, g) = − 4π(m−1)!
〈♣(c1), [ω]m−1〉+∫Msgdµg
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For an ALE Kahler manifold (M2m, g, J),
4πm(2m−1)(m−1)!
m(M, g) = − 4π(m−1)!
〈♣(c1), [ω]m−1〉+∫Msgdµg
So the mass is a “boundary correction” to the topo-logical formula for the total scalar curvature.
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We can now state our mass formula:
Theorem C. Any ALE Kahler manifold (M, g, J)of complex dimension m has mass given by
m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1
+(m− 1)!
4(2m− 1)πm
∫Msgdµg
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We can now state our mass formula:
Corollary. Any ALE scalar-flat Kahler mani-fold (M, g, J) of complex dimension m has massgiven by
m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1
.
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We can now state our mass formula:
Corollary. Any ALE scalar-flat Kahler mani-fold (M, g, J) of complex dimension m has massgiven by
m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1
.
So Theorem A is an immediate consequence!
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Rough Idea of Proof:
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Rough Idea of Proof:
Special Case: Suppose
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Rough Idea of Proof:
Special Case: Suppose
•m = 2, n = 4;
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Rough Idea of Proof:
Special Case: Suppose
•m = 2, n = 4;
• Scalar flat: s ≡ 0; and
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Rough Idea of Proof:
Special Case: Suppose
•m = 2, n = 4;
• Scalar flat: s ≡ 0; and
• Complex structure J standard at infinity:
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Rough Idea of Proof:
Special Case: Suppose
•m = 2, n = 4;
• Scalar flat: s ≡ 0; and
• Complex structure J standard at infinity:
(M −K, J) ≈bih (C2 −B4)/Γ.
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Rough Idea of Proof:
Special Case: Suppose
•m = 2, n = 4;
• Scalar flat: s ≡ 0; and
• Complex structure J standard at infinity:
(M −K, J) ≈bih (C2 −B4)/Γ.
Since g is Kahler, the complex coordinates
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Rough Idea of Proof:
Special Case: Suppose
•m = 2, n = 4;
• Scalar flat: s ≡ 0; and
• Complex structure J standard at infinity:
(M −K, J) ≈bih (C2 −B4)/Γ.
Since g is Kahler, the complex coordinates
(z1, z2) = (x1 + ix2, x3 + ix4)
are harmonic. So xj are harmonic, too, and
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Rough Idea of Proof:
Special Case: Suppose
•m = 2, n = 4;
• Scalar flat: s ≡ 0; and
• Complex structure J standard at infinity:
(M −K, J) ≈bih (C2 −B4)/Γ.
Since g is Kahler, the complex coordinates
(z1, z2) = (x1 + ix2, x3 + ix4)
are harmonic. So xj are harmonic, too, and
267
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Rough Idea of Proof:
Special Case: Suppose
•m = 2, n = 4;
• Scalar flat: s ≡ 0; and
• Complex structure J standard at infinity:
(M −K, J) ≈bih (C2 −B4)/Γ.
Since g is Kahler, the complex coordinates
(z1, z2) = (x1 + ix2, x3 + ix4)
are harmonic. So xj are harmonic, too, and
gjk(gj`,k − gjk,`
)ν`αE = −?d log
(√det g
)+O(%−3−ε).
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m(M, g) = − lim%→∞
1
12π2
∫S%/Γ
? d(
log√
det g)
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m(M, g) = − lim%→∞
1
12π2
∫S%/Γ
? d(
log√
det g)
Now set θ = i2(∂ − ∂)
(log√
det g), so that
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m(M, g) = − lim%→∞
1
12π2
∫S%/Γ
? d(
log√
det g)
Now set θ = i2(∂ − ∂)
(log√
det g), so that
ρ = dθ
is Ricci form, and
271
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m(M, g) = − lim%→∞
1
12π2
∫S%/Γ
? d(
log√
det g)
Now set θ = i2(∂ − ∂)
(log√
det g), so that
ρ = dθ
is Ricci form, and
−?d log(√
det g)
= 2 θ ∧ ω.
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m(M, g) = − lim%→∞
1
12π2
∫S%/Γ
? d(
log√
det g)
Now set θ = i2(∂ − ∂)
(log√
det g), so that
ρ = dθ
is Ricci form, and
−?d log(√
det g)
= 2 θ ∧ ω.Thus
m(M, g) = − lim%→∞
1
6π2
∫S%/Γ
θ ∧ ω
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m(M, g) = − lim%→∞
1
12π2
∫S%/Γ
? d(
log√
det g)
Now set θ = i2(∂ − ∂)
(log√
det g), so that
ρ = dθ
is Ricci form, and
−?d log(√
det g)
= 2 θ ∧ ω.Thus
m(M, g) = − lim%→∞
1
6π2
∫S%/Γ
θ ∧ ω
However, since s = 0,
d(θ ∧ ω) = ρ ∧ ω =s
4ω2 = 0.
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m(M, g) = − lim%→∞
1
12π2
∫S%/Γ
? d(
log√
det g)
Now set θ = i2(∂ − ∂)
(log√
det g), so that
ρ = dθ
is Ricci form, and
−?d log(√
det g)
= 2 θ ∧ ω.Thus
m(M, g) = − 1
6π2
∫S%/Γ
θ ∧ ω
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Let f : M → R be smooth cut-off function:
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Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
..................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
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.......
.......
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.......
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radius
1
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...........................................................................................................................................................................................................................................................................
............................................................................................
...........................................................................................
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Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
..................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.......
.......
.......
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.......
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......................
radius
1
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...........................................................................................................................................................................................................................................................................
............................................................................................
...........................................................................................
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Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
..................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
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......................
radius
1
.............................................................................................................................................................................................................................................................
...........................................................................................................................................................................................................................................................................
............................................................................................
...........................................................................................
279
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Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
..................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
......................
radius
“end”
1
.............................................................................................................................................................................................................................................................
...........................................................................................................................................................................................................................................................................
.......
.......
.......
.......
.......
.......
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.......
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.......
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.......
.......
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.......
.......
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.
..........................................................................................................................................................................................................................................................................................................................................................................................................................................
............................................................................................
...........................................................................................
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Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
..................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
......................
radius
“end”
1
.............................................................................................................................................................................................................................................................
...........................................................................................................................................................................................................................................................................
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.
..........................................................................................................................................................................................................................................................................................................................................................................................................................................
............................................................................................
...........................................................................................
...............................................................................................................................................................................................................................................................................................................................
...............................................................................................................................................................................................................................................................................................................................
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..............................................................
..................................
..................................
..................................................
.....................
...............................
281
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Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
..................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
......................
radius
1
.............................................................................................................................................................................................................................................................
...........................................................................................................................................................................................................................................................................
............................................................................................
...........................................................................................
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Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
..................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
......................
radius
M%
%
1
.............................................................................................................................................................................................................................................................
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.......
.......
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.......
.......
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..
.......
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.......
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...........................................................................................
283
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Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
..................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
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.......
.......
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.......
.......
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.......
.......
.......
.......
.......
.......
.......
.......
.......
......................
radius
M%
%
1
.............................................................................................................................................................................................................................................................
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..
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.....
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Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
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Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
Set
ψ := ρ− d(fθ)
286
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Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
Set
ψ := ρ− d(fθ)
Compactly supported, because dθ = ρ near infinity.
287
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Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
Set
ψ := ρ− d(fθ)
[ψ] = ♣([ρ]) = 2π♣(c1) ∈ H2c (M)
288
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Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
Set
ψ := ρ− d(fθ)
[ψ] = ♣([ρ]) = 2π♣(c1) ∈ H2c (M)
〈2π♣(c1), [ω]〉 =
∫Mψ ∧ ω
289
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Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
Set
ψ := ρ− d(fθ)
[ψ] = ♣([ρ]) = 2π♣(c1) ∈ H2c (M)
〈2π♣(c1), [ω]〉 =
∫M%
ψ ∧ ω
where M% defined by radius ≤ %.
290
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Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
Set
ψ := ρ− d(fθ)
[ψ] = ♣([ρ]) = 2π♣(c1) ∈ H2c (M)
〈2π♣(c1), [ω]〉 =
∫M%
ψ ∧ ω
291
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Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
Set
ψ := ρ− d(fθ)
[ψ] = ♣([ρ]) = 2π♣(c1) ∈ H2c (M)
〈2π♣(c1), ω〉 =
∫M%
[ρ− d(fθ)] ∧ ω
292
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Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
Set
ψ := ρ− d(fθ)
[ψ] = ♣([ρ]) = 2π♣(c1) ∈ H2c (M)
〈2π♣(c1), ω〉 =
∫M%
[ρ− d(fθ)] ∧ ω
because scalar-flat =⇒ ρ ∧ ω = 0.
293
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Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
Set
ψ := ρ− d(fθ)
[ψ] = ♣([ρ]) = 2π♣(c1) ∈ H2c (M)
〈2π♣(c1), [ω]〉 = −∫M%
d(fθ) ∧ ω
because scalar-flat =⇒ ρ ∧ ω = 0.
294
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Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
Set
ψ := ρ− d(fθ)
[ψ] = ♣([ρ]) = 2π♣(c1) ∈ H2c (M)
〈2π♣(c1), [ω]〉 = −∫M%
d(fθ ∧ ω)
because scalar-flat =⇒ ρ ∧ ω = 0.
295
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Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
Set
ψ := ρ− d(fθ)
[ψ] = ♣([ρ]) = 2π♣(c1) ∈ H2c (M)
〈2π♣(c1), [ω]〉 = −∫M%
d(fθ ∧ ω)
296
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Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
Set
ψ := ρ− d(fθ)
[ψ] = ♣([ρ]) = 2π♣(c1) ∈ H2c (M)
〈2π♣(c1), [ω]〉 = −∫∂M%
fθ ∧ ω
by Stokes’ theorem.
297
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Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
Set
ψ := ρ− d(fθ)
[ψ] = ♣([ρ]) = 2π♣(c1) ∈ H2c (M)
〈2π♣(c1), [ω]〉 = −∫∂M%
θ ∧ ω
by Stokes’ theorem.
298
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Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
Set
ψ := ρ− d(fθ)
[ψ] = ♣([ρ]) = 2π♣(c1) ∈ H2c (M)
〈2π♣(c1), [ω]〉 = −∫S%/Γ
θ ∧ ω
by Stokes’ theorem.
299
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Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
Set
ψ := ρ− d(fθ)
[ψ] = ♣([ρ]) = 2π♣(c1) ∈ H2c (M)
〈2π♣(c1), [ω]〉 = −∫S%/Γ
θ ∧ ω
by Stokes’ theorem.
300
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Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
Set
ψ := ρ− d(fθ)
[ψ] = ♣([ρ]) = 2π♣(c1) ∈ H2c (M)
〈2π♣(c1), [ω]〉 = −∫S%/Γ
θ ∧ ω
by Stokes’ theorem.
So
m(M, g) = − 1
6π2
∫S%/Γ
θ ∧ ω
301
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Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
Set
ψ := ρ− d(fθ)
[ψ] = ♣([ρ]) = 2π♣(c1) ∈ H2c (M)
〈2π♣(c1), [ω]〉 = −∫S%/Γ
θ ∧ ω
by Stokes’ theorem.
So
m(M, g) = − 1
3π〈♣(c1), [ω]〉
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Let f : M → R be smooth cut-off function:≡ 0 away from end,≡ 1 near infinity.
Set
ψ := ρ− d(fθ)
[ψ] = ♣([ρ]) = 2π♣(c1) ∈ H2c (M)
〈2π♣(c1), [ω]〉 = −∫S%/Γ
θ ∧ ω
by Stokes’ theorem.
So
m(M, g) = − 1
3π〈♣(c1), [ω]〉
as claimed.
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We assumed:
304
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We assumed:
•m = 2;
• s ≡ 0; and
• Complex structure J standard at infinity.
305
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General case:
306
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General case:
•General m ≥ 2: straightforward. . .
307
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General case:
•General m ≥ 2: straightforward. . .
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General case:
•General m ≥ 2: straightforward. . .
• s 6≡ 0, compensate by adding∫s dµ. . .
309
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General case:
•General m ≥ 2: straightforward. . .
• s 6≡ 0, compensate by adding∫s dµ. . .
• If m > 2, J is always standard at infinity.
310
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General case:
•General m ≥ 2: straightforward. . .
• s 6≡ 0, compensate by adding∫s dµ. . .
• If m > 2, J is always standard at infinity.
• If m = 2 and AE, J is still standard at infinity.
311
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General case:
•General m ≥ 2: straightforward. . .
• s 6≡ 0, compensate by adding∫s dµ. . .
• If m > 2, J is always standard at infinity.
• If m = 2 and AE, J is still standard at infinity.
• If m = 2 and ALE, J can be non-standard at∞.
312
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General case:
•General m ≥ 2: straightforward. . .
• s 6≡ 0, compensate by adding∫s dµ. . .
• If m > 2, J is always standard at infinity.
• If m = 2 and AE, J is still standard at infinity.
• If m = 2 and ALE, J can be non-standard at∞.
The last point is serious.
313
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General case:
•General m ≥ 2: straightforward. . .
• s 6≡ 0, compensate by adding∫s dµ. . .
• If m > 2, J is always standard at infinity.
• If m = 2 and AE, J is still standard at infinity.
• If m = 2 and ALE, J can be non-standard at∞.
Seen in “gravitational instantons”
and other explicit examples.
314
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General case:
•General m ≥ 2: straightforward. . .
• s 6≡ 0, compensate by adding∫s dµ. . .
• If m > 2, J is always standard at infinity.
• If m = 2 and AE, J is still standard at infinity.
• If m = 2 and ALE, J can be non-standard at∞.
One argument proceeds by osculation:
315
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General case:
•General m ≥ 2: straightforward. . .
• s 6≡ 0, compensate by adding∫s dµ. . .
• If m > 2, J is always standard at infinity.
• If m = 2 and AE, J is still standard at infinity.
• If m = 2 and ALE, J can be non-standard at∞.
One argument proceeds by osculation:
J = J0 + O(%−3), OJ = O(%−4)
in suitable asymptotic coordinates adapted to g.
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To understand J at infinity:
317
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To understand J at infinity:
Let M∞ be universal over of end M∞.
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To understand J at infinity:
Let M∞ be universal over of end M∞.
Cap off M∞ by adding CPm−1 at infinity.
319
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To understand J at infinity:
Let M∞ be universal over of end M∞.
Cap off M∞ by adding CPm−1 at infinity.
Added hypersurface CPm−1 has normal bundleO(1).
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To understand J at infinity:
Let M∞ be universal over of end M∞.
Cap off M∞ by adding CPm−1 at infinity.
Added hypersurface CPm−1 has normal bundleO(1).
Complete analytic family encodes info about J .
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To understand J at infinity:
322
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To understand J at infinity:
AE case:
Compactify M itself by adding CPm−1 at infinity.
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To understand J at infinity:
AE case:
Compactify M itself by adding CPm−1 at infinity.
Linear system of CPm−1 gives holomorphic map
(M ∪ CPm−1)→ CPmwhich is biholomorphism near CPm−1.
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To understand J at infinity:
AE case:
Compactify M itself by adding CPm−1 at infinity.
Linear system of CPm−1 gives holomorphic map
(M ∪ CPm−1)→ CPmwhich is biholomorphism near CPm−1.
Thus obtain holomorphic map
Φ : M → Cm
which is biholomorphism near infinity.
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To understand J at infinity:
AE case:
Compactify M itself by adding CPm−1 at infinity.
Linear system of CPm−1 gives holomorphic map
(M ∪ CPm−1)→ CPmwhich is biholomorphism near CPm−1.
Thus obtain holomorphic map
Φ : M → Cm
which is biholomorphism near infinity.
This has some interesting consequences. . .
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Theorem D (Positive Mass Theorem). Any AEKahler manifold with non-negative scalar curva-ture has non-negative mass:
327
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Theorem D (Positive Mass Theorem). Any AEKahler manifold with non-negative scalar curva-ture has non-negative mass:
328
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Theorem D (Positive Mass Theorem). Any AEKahler manifold with non-negative scalar curva-ture has non-negative mass:
329
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Theorem D (Positive Mass Theorem). Any AEKahler manifold with non-negative scalar curva-ture has non-negative mass:
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Theorem D (Positive Mass Theorem). Any AEKahler manifold with non-negative scalar curva-ture has non-negative mass:
AE & Kahler & s ≥ 0 =⇒ m(M, g) ≥ 0.
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Theorem D (Positive Mass Theorem). Any AEKahler manifold with non-negative scalar curva-ture has non-negative mass:
AE & Kahler & s ≥ 0 =⇒ m(M, g) ≥ 0.
Moreover, m = 0 ⇐⇒
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Theorem D (Positive Mass Theorem). Any AEKahler manifold with non-negative scalar curva-ture has non-negative mass:
AE & Kahler & s ≥ 0 =⇒ m(M, g) ≥ 0.
Moreover, m = 0⇐⇒ (M, g) is Euclidean space.
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Theorem D (Positive Mass Theorem). Any AEKahler manifold with non-negative scalar curva-ture has non-negative mass:
AE & Kahler & s ≥ 0 =⇒ m(M, g) ≥ 0.
Moreover, m = 0⇐⇒ (M, g) is Euclidean space.
Proof actually shows something stronger!
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Theorem E (Penrose Inequality). Let (M, g, J)be an AE Kahler manifold with scalar curvatures ≥ 0. Then (M,J) carries a canonical divisorD that is expressed as a sum
∑jnjDj of com-
pact complex hypersurfaces with positive integercoefficients, with the property that
⋃jDj 6= ∅
whenever (M,J) 6= Cm. In terms of this divi-sor, we then have
335
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Theorem E (Penrose Inequality). Let (M2m, g, J)be an AE Kahler manifold with scalar curvatures ≥ 0. Then (M,J) carries a canonical divisorD that is expressed as a sum
∑jnjDj of com-
pact complex hypersurfaces with positive integercoefficients, with the property that
⋃jDj 6= ∅
whenever (M,J) 6= Cm. In terms of this divi-sor, we then have
336
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Theorem E (Penrose Inequality). Let (M2m, g, J)be an AE Kahler manifold with scalar curvatures ≥ 0. Then (M,J) carries a canonical divisorD that is expressed as a sum
∑jnjDj of com-
pact complex hypersurfaces with positive integercoefficients, with the property that
⋃jDj 6= ∅
whenever (M,J) 6= Cm. In terms of this divi-sor, we then have
337
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Theorem E (Penrose Inequality). Let (M2m, g, J)be an AE Kahler manifold with scalar curvatures ≥ 0. Then (M,J) carries a canonical divisorD that is expressed as a sum
∑jnjDj of com-
pact complex hypersurfaces with positive integercoefficients, with the property that
⋃jDj 6= ∅
whenever (M,J) 6= Cm. In terms of this divi-sor, we then have
338
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Theorem E (Penrose Inequality). Let (M2m, g, J)be an AE Kahler manifold with scalar curvatures ≥ 0. Then (M,J) carries a canonical divisorD that is expressed as a sum
∑jnjDj of com-
pact complex hypersurfaces with positive integercoefficients, with the property that
⋃jDj 6= ∅
whenever (M,J) 6= Cm. In terms of this divi-sor, we then have
339
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Theorem E (Penrose Inequality). Let (M2m, g, J)be an AE Kahler manifold with scalar curvatures ≥ 0. Then (M,J) carries a canonical divisorD that is expressed as a sum
∑jnjDj of com-
pact complex hypersurfaces with positive integercoefficients, with the property that
⋃jDj 6= ∅
whenever (M,J) 6= Cm. In terms of this divi-sor, we then have
340
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Theorem E (Penrose Inequality). Let (M2m, g, J)be an AE Kahler manifold with scalar curvatures ≥ 0. Then (M,J) carries a canonical divisorD that is expressed as a sum
∑jnjDj of com-
pact complex hypersurfaces with positive integercoefficients, with the property that
⋃jDj 6= ∅
whenever (M,J) 6= Cm. In terms of this divi-sor, we then have
341
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Theorem E (Penrose Inequality). Let (M2m, g, J)be an AE Kahler manifold with scalar curvatures ≥ 0. Then (M,J) carries a canonical divisorD that is expressed as a sum
∑jnjDj of com-
pact complex hypersurfaces with positive integercoefficients, with the property that
⋃jDj 6= ∅
whenever (M,J) 6= Cm. In terms of this divi-sor, we then have
342
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Theorem E (Penrose Inequality). Let (M2m, g, J)be an AE Kahler manifold with scalar curvatures ≥ 0. Then (M,J) carries a canonical divisorD that is expressed as a sum
∑jnjDj of com-
pact complex hypersurfaces with positive integercoefficients, with the property that
⋃jDj 6= ∅
whenever (M,J) 6= Cm. In terms of this divi-sor, we then have
343
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Theorem E (Penrose Inequality). Let (M2m, g, J)be an AE Kahler manifold with scalar curvatures ≥ 0. Then (M,J) carries a canonical divisorD that is expressed as a sum
∑jnjDj of com-
pact complex hypersurfaces with positive integercoefficients, with the property that
⋃jDj 6= ∅
whenever (M,J) 6= Cm. In terms of this divi-sor, we then have
m(M, g) ≥ (m− 1)!
(2m− 1)πm−1
∑njVol (Dj)
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Theorem E (Penrose Inequality). Let (M2m, g, J)be an AE Kahler manifold with scalar curvatures ≥ 0. Then (M,J) carries a canonical divisorD that is expressed as a sum
∑jnjDj of com-
pact complex hypersurfaces with positive integercoefficients, with the property that
⋃jDj 6= ∅
whenever (M,J) 6= Cm. In terms of this divi-sor, we then have
m(M, g) ≥ (m− 1)!
(2m− 1)πm−1
∑njVol (Dj)
345
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Theorem E (Penrose Inequality). Let (M2m, g, J)be an AE Kahler manifold with scalar curvatures ≥ 0. Then (M,J) carries a canonical divisorD that is expressed as a sum
∑jnjDj of com-
pact complex hypersurfaces with positive integercoefficients, with the property that
⋃jDj 6= ∅
whenever (M,J) 6= Cm. In terms of this divi-sor, we then have
m(M, g) ≥ (m− 1)!
(2m− 1)πm−1
∑njVol (Dj)
346
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Theorem E (Penrose Inequality). Let (M2m, g, J)be an AE Kahler manifold with scalar curvatures ≥ 0. Then (M,J) carries a canonical divisorD that is expressed as a sum
∑jnjDj of com-
pact complex hypersurfaces with positive integercoefficients, with the property that
⋃jDj 6= ∅
whenever (M,J) 6= Cm. In terms of this divi-sor, we then have
m(M, g) ≥ (m− 1)!
(2m− 1)πm−1
∑njVol (Dj)
with = ⇐⇒
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Theorem E (Penrose Inequality). Let (M2m, g, J)be an AE Kahler manifold with scalar curvatures ≥ 0. Then (M,J) carries a canonical divisorD that is expressed as a sum
∑jnjDj of com-
pact complex hypersurfaces with positive integercoefficients, with the property that
⋃jDj 6= ∅
whenever (M,J) 6= Cm. In terms of this divi-sor, we then have
m(M, g) ≥ (m− 1)!
(2m− 1)πm−1
∑njVol (Dj)
with = ⇐⇒ (M, g, J) is scalar-flat Kahler.
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This follows from existence of a holomorphic map
Φ : M → Cm
which is a biholomorphism near infinity.
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This follows from existence of a holomorphic map
Φ : M → Cm
which is a biholomorphism near infinity.
Indeed, we then have a holomorphic section
ϕ = Φ∗dz1 ∧ · · · ∧ dzm
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This follows from existence of a holomorphic map
Φ : M → Cm
which is a biholomorphism near infinity.
Indeed, we then have a holomorphic section
ϕ = Φ∗dz1 ∧ · · · ∧ dzm
of the canonical line bundle which vanishes exactlyat the critical points of Φ.
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This follows from existence of a holomorphic map
Φ : M → Cm
which is a biholomorphism near infinity.
Indeed, we then have a holomorphic section
ϕ = Φ∗dz1 ∧ · · · ∧ dzm
of the canonical line bundle which vanishes exactlyat the critical points of Φ.
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This follows from existence of a holomorphic map
Φ : M → Cm
which is a biholomorphism near infinity.
Indeed, we then have a holomorphic section
ϕ = Φ∗dz1 ∧ · · · ∧ dzm
of the canonical line bundle which vanishes exactlyat the critical points of Φ.
The zero set of ϕ, counted with multiplicities, givesus a canonical divisor
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This follows from existence of a holomorphic map
Φ : M → Cm
which is a biholomorphism near infinity.
Indeed, we then have a holomorphic section
ϕ = Φ∗dz1 ∧ · · · ∧ dzm
of the canonical line bundle which vanishes exactlyat the critical points of Φ.
The zero set of ϕ, counted with multiplicities, givesus a canonical divisor
D =∑
njDj
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This follows from existence of a holomorphic map
Φ : M → Cm
which is a biholomorphism near infinity.
Indeed, we then have a holomorphic section
ϕ = Φ∗dz1 ∧ · · · ∧ dzm
of the canonical line bundle which vanishes exactlyat the critical points of Φ.
The zero set of ϕ, counted with multiplicities, givesus a canonical divisor
D =∑
njDj
and
−〈♣(c1),ωm−1
(m− 1)!〉 =
∑njVol (Dj)
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This follows from existence of a holomorphic map
Φ : M → Cm
which is a biholomorphism near infinity.
Indeed, we then have a holomorphic section
ϕ = Φ∗dz1 ∧ · · · ∧ dzm
of the canonical line bundle which vanishes exactlyat the critical points of Φ.
The zero set of ϕ, counted with multiplicities, givesus a canonical divisor
D =∑
njDj
and
−〈♣(c1),ωm−1
(m− 1)!〉 =
∑njVol (Dj)
so the mass formula implies the claim.
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m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1
+(m− 1)!
4(2m− 1)πm
∫Msgdµg
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m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1
+(m− 1)!
4(2m− 1)πm
∫Msgdµg
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m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1
+(m− 1)!
4(2m− 1)πm
∫Msgdµg
...............................................................................................................................................................................................................................................................................................................................
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m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1
+(m− 1)!
4(2m− 1)πm
∫Msgdµg
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m(M, g) = −〈♣(c1), [ω]m−1〉(2m− 1)πm−1
+(m− 1)!
4(2m− 1)πm
∫Msgdµg
...............................................................................................................................................................................................................................................................................................................................
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362
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Happy Birthday, Roger!
363
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Happy Birthday, Roger!
Happy Birthday, Twistors!
364
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Happy Birthday, Roger!
Happy Birthday, Twistors!
Happy Non-Retirement, Nick!
365