Geometry - uni-muenster.de
Transcript of Geometry - uni-muenster.de
iins er berseminar
> ifferenial Geometry
n iy , Juno-
,2 2
Cp Convergence and e- Zeguariy he rems
fr ntropy and Scalar Curvature wer Bounds
' '
n-
w an- Chun ee and -
ar n aber
heme :
M"
. g) Riemannian manifold.
Put some constraints What structure does
on the curvature~
this impose on ( M. g) ?
to study :
lake sequence(Mi
, g;) satisfying constraint , look at limit-
Mi >
I 1-
What structure does have ?In what sense ?
F- ✗ Spaces with Ricci lower bounds
( Mi, g;) sat
.Ric ; 3
- IgiGH
( Mi , di ) → ( ✗ , d) Cheeger - Codding .
E - Regularity 1hm ( C '
ng'
97, Creager - C
'
ng'
97
tix E > O. there exists of = die
,n) > 0 such that if
( M. g) Riemannian manifold .
x c- M w/
Ricg 3 -
org volgl BC x. l)) 3 ( I - 8) Wn
'
henoc. " ( Bix,
D, 1310"
.
I ) ) s E.
His
In fact,GH can be
improved to biHolder
homeomorphic by iteratingon all scales .
to summarize :
Almost Euclidean nc apsing- Im s
-
Ricci cower boundt
uclidean volume
Geometrically close
to Euclidean
ur goal today :
E regularity assuming only scalar curvature lower bounds.
- mos- uclidean "l oncollapsing
"- Almost
scaar I wer s una Euclidean Perelman
entropy
Rg ? - d
"
Geometrically close < Not in any metric
0 uclidean" Space sense !
In"
dp sense."
Background Perelman Entropy µ g. t ) . T> 0
Optimal constant in family of log- Sobolev inequalities .
Def
(a) u g. I inf{ Ñlu) : /µ uidvolg = }
Ñlu = fm 4 rut + Rgu'
n' Iogu'
dvol.gr I log'll
if : og Sobolev on Euclidean space ⇐ ul gens , 1)= 0
.
fuilogu' s / 410nF dx n + log (4-11)-% if Ju' =L .
wi-h =L v2 = ( 4Ñ%e×p{ - ✗☐
2a }
t(b) vulg , L) = je T -1g ,
I
Some key perspective notes
M . g) complete with bounded curvature.
•• u( g. T ) ± for all T > 0
with equality < > M. g) = R
"
, gene
• Mlg ,T ) = - d for TE ,
I, 3g ? d
g Bglxir ? I - E) r "wn
re 0.1 ].
We will assume
µ / g. T 3 - d for TE 0,I
I w e 's turn to "
Geometrically close
understanding "
o uclidean
ok at limits of sequences ( Mi , g;) satisfying✓
Rg ;> '
i, m gi . L
) Z - Yi -
Lc- (O
. I
n what sense do they converge-
o IR? geuc?
A key point :
heir metric space structures
need not converge!
in: 3asic Idea :
' etric go on 2"
2" '
x 3
d×zEdx '
*A fibers
degenerateSaurer slowlyfrom one scale to
next
• ••.
• (IR" ", gene
r - I r = I
•
Metric is Euclidean
out here.
go
*
a ro
Can make this change
sow enough such that
IR
° Rage - de ) ' O, a
white lie : o
get scalar lower sound ,we need iery flat
• JU Ge , L ) - CCE ) s O cone me ric'
on 2h- '
n > 4.
T Te ( O,I
3.u n the pointed G - innit,centra line co apses to a point.
and
Voge Bge .
'
e : After rescaling chopping -action
, we ge
2 :
ake a flat ones I ? g. flat ①
2 btain ! gi by pasting
increasingly cense coaction f disjointostrips along parallel copies of S!
3 Can choose parameters such that
Rg ; > -'i µ gi . L 3 of t TECO
,I
3. ul
Gromov - Hausdorff ④G1
limit is 1" ! goat
intrinsic flat imi0 current
.
is zero Curren
' "
ake a flat oras in. go.at
÷÷÷÷÷i÷÷÷÷÷÷÷÷÷÷÷÷:÷¥E¥#¥E3 Can choose parameters such that
Rg ; > -'i ju gi . L 3 do F Te (O
,I
3. ul
c
: ¥¥¥¥#¥ ... .
#¥¥¥#¥ o ...
Up Sho : under scalar curva are and entropy cower bounds,
cis ance functions can
,
behave very poorly .
PER, p>
newnotion :
cp distance .
dlx.gl = sup { fix)- fly) : 1199-16<-1 }
Jef : M.gl Riemannian manifold. x.ge M, P > n
.
-
a) dplx . y) sup { Ifcx ) fly) / : f. left " dvolg ± I } .
b Bp × .r ) = { ye M
: dplx,y< r}
✓ FkFk → I a. e. fk → 1 in LP
.
£fk ↳ 1 in tf
some key perspective no-
es
• dolx.gl = dlx.gl and Iimp→ ,
dplxiy ) = dlx.glI
• dp reflects behavior of" P Sobolev space
• On euclidean space .- Mp
dplx.gl Spin ✗ - y .
• M, g) ,
(Nih) Cpt Riemannian manifolds .
If (M.dp.gr) , N, dp.in) isometric as metric spaces
then (M . g) ,( N
,h) isometric . as Riem . mflds .
1hm (-ee - Naber -N.
'
20)
tix n >_ 2, p ? n -11
,E > 0
.There exists D= dcn.p.ae) such that if
Mn, g complete with bounded curvature with
Ray Z - d, Mlg .
c) Z - J H Te 10.1-
then for any ✗ e-M,
E,•
CGH ( Bp.gl/.l,dp.g,Bp.geuc 0"
,I
, pieuc
V0 /g Bp.g ×
,r
• E S + E
Vgene(Bpigeuc 0 .
r )it recoil) .
1hm (-ee - Naber -N.
'
20)
Fix n > 2, oft 1
,E > 0
.There exists D= dcn
, ,e) such that if-
M"
, g closed with
Ray Z - d, vulg .
-43 - S H Te 10.1-
then
fml Rg 't
dvolg S E
Schoen Yau : if ( T? g) has Rg 30 ,then
g= gflat .
1hm (-ee - Naber -N.
'
20)
tix n >_ 2, p ? n -11
,> 0
.here exists D= dln ,p s
.
-
.if ?g ;)
satisfy
Rgi ? -'i, µ(gi.tl?-dV-- c- (0,1 ]
, Volgi(Mi s.
'
then up to subsequence,
Mi. gi
" ( T". gfiat in dp sense .
Limit spaces are \ metric spaces , they are
Rectifiable Ziemann ian spaces. g
I , m topological space with a Borel measure, equipped with
• Atlas of charts with bi Lipschitz transition
maps covering up to set of measure zero.
•• Possibly singular metric g defined in charts.
This is enough structure
to define W " ?
1hm (-ee - Naber -N.
'
20)
tix n > 2, p s n -11
.there exists or = den
, p) such that if { Mi, g ; , Xi
}
complete with bounded curvature with
Rg ;3 - d
, dig i. T) Z - J V te (0,1
Some subtlety tothen up to subsequence,
defining this
I i . gi , Xi" X
, g .x) in pointed dp sense a
where, g. x is a rectifiabe Riemannian space that is
• W" " - rectifiable complete
( W " "
space is"
big" and " well - behaved
"
• dp - complete
dp is metric generating same topology as
• is a smooth topological manifold .
u !