Closed magnetic geodesics on Riemann surfaces
Transcript of Closed magnetic geodesics on Riemann surfaces
Closed magnetic geodesics on Riemann surfaces
Matthias Schneider
Mathematisches InstitutUniversity of Heidelberg
April 8. 2011, Sevilla
The problemGiven
I an oriented (compact) surface (M, g) with a Riemannianmetric g ,
I a smooth (positive) function k : M → R.
Problem: Existence of a closed immersed curve γ : S1 → M withgeodesic curvature kg (γ, t) = k(γ(t)).
I Geodesic curvature:kg (γ, t) := |γ(t)|−3g 〈Dt,g γ(t), Jg (γ(t))γ(t)〉,Jg (p): rotation by +π/2 in TpM w.r.t. the given orientationand metric.
I These curves are called magnetic geodesics. Theycorrespond to trajectories of a charged particle on M in amagnetic field with magnetic form kdµg and solve
Dt,g γ = |γ|gk(γ)Jg (γ)γ.
The problemGiven
I an oriented (compact) surface (M, g) with a Riemannianmetric g ,
I a smooth (positive) function k : M → R.
Problem: Existence of a closed immersed curve γ : S1 → M withgeodesic curvature kg (γ, t) = k(γ(t)).
I Geodesic curvature:kg (γ, t) := |γ(t)|−3g 〈Dt,g γ(t), Jg (γ(t))γ(t)〉,Jg (p): rotation by +π/2 in TpM w.r.t. the given orientationand metric.
I These curves are called magnetic geodesics. Theycorrespond to trajectories of a charged particle on M in amagnetic field with magnetic form kdµg and solve
Dt,g γ = |γ|gk(γ)Jg (γ)γ.
The problemGiven
I an oriented (compact) surface (M, g) with a Riemannianmetric g ,
I a smooth (positive) function k : M → R.
Problem: Existence of a closed immersed curve γ : S1 → M withgeodesic curvature kg (γ, t) = k(γ(t)).
I Geodesic curvature:kg (γ, t) := |γ(t)|−3g 〈Dt,g γ(t), Jg (γ(t))γ(t)〉,Jg (p): rotation by +π/2 in TpM w.r.t. the given orientationand metric.
I These curves are called magnetic geodesics. Theycorrespond to trajectories of a charged particle on M in amagnetic field with magnetic form kdµg and solve
Dt,g γ = |γ|gk(γ)Jg (γ)γ.
The spherical geometry
Example 1: The round sphere (S2, g0) with k ≡ k0.
I S2 = ∂B1(0) ⊂ R3
I g0 induced metric
I Gauss curvatureKg0 ≡ 1
I Magnetic geodesicswith k0 = 1.
The spherical geometry
Example 1: The round sphere (S2, g0) with k ≡ k0.
I S2 = ∂B1(0) ⊂ R3
I g0 induced metric
I Gauss curvatureKg0 ≡ 1
I Magnetic geodesicswith k0 = 1.
The spherical geometry
Example 1: The round sphere (S2, g0) with k ≡ k0.
I S2 = ∂B1(0) ⊂ R3
I g0 induced metric
I Gauss curvatureKg0 ≡ 1
I Magnetic geodesicswith k0 = 1.
The spherical geometry
Example 1: The round sphere (S2, g0) with k ≡ k0.
I S2 = ∂B1(0) ⊂ R3
I g0 induced metric
I Gauss curvatureKg0 ≡ 1
I Magnetic geodesicswith k0 = 1.
The hyperbolic geometry
Example 2: The hyperbolic plane (H, gH) with k ≡ k0.
I H = B1(0) ⊂ R2
I gH =4(1− |x |2)−2gR2
I Gauss curvatureKgH ≡ −1
I Magnetic geodesicsfor k0 = 2
The hyperbolic geometry
Example 2: The hyperbolic plane (H, gH) with k ≡ k0.
I H = B1(0) ⊂ R2
I gH =4(1− |x |2)−2gR2
I Gauss curvatureKgH ≡ −1
I Magnetic geodesicsfor k0 = 2
The hyperbolic geometry
Example 2: The hyperbolic plane (H, gH) with k ≡ k0.
I H = B1(0) ⊂ R2
I gH =4(1− |x |2)−2gR2
I Gauss curvatureKgH ≡ −1
I Magnetic geodesicsfor k0 = 2
The hyperbolic geometry
Example 2: The hyperbolic plane (H, gH) with k ≡ k0.
I H = B1(0) ⊂ R2
I gH =4(1− |x |2)−2gR2
I Gauss curvatureKgH ≡ −1
I Magnetic geodesicsfor k0 = 2
The hyperbolic geometry
Example 2: The hyperbolic plane (H, gH) with k ≡ k0.
I H = B1(0) ⊂ R2
I gH =4(1− |x |2)−2gR2
I Gauss curvatureKgH ≡ −1
I Magnetic geodesicsfor k0 = 2
The hyperbolic geometry
Example 2: The hyperbolic plane (H, gH) with k ≡ k0.
I H = B1(0) ⊂ R2
I gH =4(1− |x |2)−2gR2
I Gauss curvatureKgH ≡ −1
I Magnetic geodesicsfor k0 = 2
Methods and approaches
There is a vast literature on the existence of closed magneticgeodesics.
I Theory of dynamical systems: Arnold ’86, Ginzburg ’96Magnetic geodesics are periodic orbits of a twistedHamiltonian flow withH = 1
2 |q|2, twisted symplectic form ω = −dλ+ π∗(kµg ),
π : TM → M, µg volume form, λ canonical 1-form (qdq).
I Morse-Novikov theory: Novikov ’84, Taimanov ’92Minimize E (γ) := 1
2
∫S1 |γ|2 + ”
∫B kµg” where ∂B = γ.
If dθ = kµg , then ”∫B kµg” =
∫γ θ.
In general, the functional E is multi-valued.
I Aubry-Mather-Theorie: Contreras, Macarini, Paternain ’04Mane’s critical value, existence on compact surfaces, if kµg isexact.
Methods and approaches
There is a vast literature on the existence of closed magneticgeodesics.
I Theory of dynamical systems: Arnold ’86, Ginzburg ’96Magnetic geodesics are periodic orbits of a twistedHamiltonian flow withH = 1
2 |q|2, twisted symplectic form ω = −dλ+ π∗(kµg ),
π : TM → M, µg volume form, λ canonical 1-form (qdq).
I Morse-Novikov theory: Novikov ’84, Taimanov ’92Minimize E (γ) := 1
2
∫S1 |γ|2 + ”
∫B kµg” where ∂B = γ.
If dθ = kµg , then ”∫B kµg” =
∫γ θ.
In general, the functional E is multi-valued.
I Aubry-Mather-Theorie: Contreras, Macarini, Paternain ’04Mane’s critical value, existence on compact surfaces, if kµg isexact.
Methods and approaches
There is a vast literature on the existence of closed magneticgeodesics.
I Theory of dynamical systems: Arnold ’86, Ginzburg ’96Magnetic geodesics are periodic orbits of a twistedHamiltonian flow withH = 1
2 |q|2, twisted symplectic form ω = −dλ+ π∗(kµg ),
π : TM → M, µg volume form, λ canonical 1-form (qdq).
I Morse-Novikov theory: Novikov ’84, Taimanov ’92Minimize E (γ) := 1
2
∫S1 |γ|2 + ”
∫B kµg” where ∂B = γ.
If dθ = kµg , then ”∫B kµg” =
∫γ θ.
In general, the functional E is multi-valued.
I Aubry-Mather-Theorie: Contreras, Macarini, Paternain ’04Mane’s critical value, existence on compact surfaces, if kµg isexact.
Methods and approaches
There is a vast literature on the existence of closed magneticgeodesics.
I Theory of dynamical systems: Arnold ’86, Ginzburg ’96Magnetic geodesics are periodic orbits of a twistedHamiltonian flow withH = 1
2 |q|2, twisted symplectic form ω = −dλ+ π∗(kµg ),
π : TM → M, µg volume form, λ canonical 1-form (qdq).
I Morse-Novikov theory: Novikov ’84, Taimanov ’92Minimize E (γ) := 1
2
∫S1 |γ|2 + ”
∫B kµg” where ∂B = γ.
If dθ = kµg , then ”∫B kµg” =
∫γ θ.
In general, the functional E is multi-valued.
I Aubry-Mather-Theorie: Contreras, Macarini, Paternain ’04Mane’s critical value, existence on compact surfaces, if kµg isexact.
Open problems
I Conjecture (*1): (Arnold ’81)If (M, g) is compact oriented surface and k is positive, thenthere is a closed k-magnetic geodesic.More precisely, there are at least two for M = S2 and at leastthree in all other cases.
I Arnold ’84: (*1) is true for a flat torus (T 2, g0).I Ginzburg ’96: (*1) is true for (M, g), if k >>> 1.
existence, if 0 < k <<< 1.I (*1) is wrong for (H/Γ, g0) with Kg0 ≡ −1 and k ≡ 1.
”Horocycle flow”, Hedlund ’36, Ginzburg ’96
I Conjecture (*2): Novikov ’82, Rosenberg and Smith ’10On (S2, g) with k positive, there is an embedded closedk-magnetic geodesic.
Open problems
I Conjecture (*1): (Arnold ’81)If (M, g) is compact oriented surface and k is positive, thenthere is a closed k-magnetic geodesic.More precisely, there are at least two for M = S2 and at leastthree in all other cases.
I Arnold ’84: (*1) is true for a flat torus (T 2, g0).I Ginzburg ’96: (*1) is true for (M, g), if k >>> 1.
existence, if 0 < k <<< 1.I (*1) is wrong for (H/Γ, g0) with Kg0 ≡ −1 and k ≡ 1.
”Horocycle flow”, Hedlund ’36, Ginzburg ’96
I Conjecture (*2): Novikov ’82, Rosenberg and Smith ’10On (S2, g) with k positive, there is an embedded closedk-magnetic geodesic.
Open problems
I Conjecture (*1): (Arnold ’81)If (M, g) is compact oriented surface and k is positive, thenthere is a closed k-magnetic geodesic.More precisely, there are at least two for M = S2 and at leastthree in all other cases.
I Arnold ’84: (*1) is true for a flat torus (T 2, g0).I Ginzburg ’96: (*1) is true for (M, g), if k >>> 1.
existence, if 0 < k <<< 1.I (*1) is wrong for (H/Γ, g0) with Kg0 ≡ −1 and k ≡ 1.
”Horocycle flow”, Hedlund ’36, Ginzburg ’96
I Conjecture (*2): Novikov ’82, Rosenberg and Smith ’10On (S2, g) with k positive, there is an embedded closedk-magnetic geodesic.
Results
I (*1) is true for (S2, g), if Kg ≥ 0 and k > 0, i.e. there are twoclosed k-magnetic geodesics in this case. (S. ’10)
I There is a closed k-magnetic geodesic, if χ(M) < 0, Kg ≥ −1and k > 1. (S. ’10)
I (*2) holds for (S2, g), i.e. there are two closed embeddedk-magnetic geodesics, under each of the following threeassumptions:
1. k > 0 and g is 14 -pinched (sup Kg < 4 inf Kg ), (S. ’09)
2. k is large enough depending on g . (S. ’09)
3. Kg > 0 and k is small enough depending on g . (Rosenberg &S. ’11)
Results
I (*1) is true for (S2, g), if Kg ≥ 0 and k > 0, i.e. there are twoclosed k-magnetic geodesics in this case. (S. ’10)
I There is a closed k-magnetic geodesic, if χ(M) < 0, Kg ≥ −1and k > 1. (S. ’10)
I (*2) holds for (S2, g), i.e. there are two closed embeddedk-magnetic geodesics, under each of the following threeassumptions:
1. k > 0 and g is 14 -pinched (sup Kg < 4 inf Kg ), (S. ’09)
2. k is large enough depending on g . (S. ’09)
3. Kg > 0 and k is small enough depending on g . (Rosenberg &S. ’11)
Results
I (*1) is true for (S2, g), if Kg ≥ 0 and k > 0, i.e. there are twoclosed k-magnetic geodesics in this case. (S. ’10)
I There is a closed k-magnetic geodesic, if χ(M) < 0, Kg ≥ −1and k > 1. (S. ’10)
I (*2) holds for (S2, g), i.e. there are two closed embeddedk-magnetic geodesics, under each of the following threeassumptions:
1. k > 0 and g is 14 -pinched (sup Kg < 4 inf Kg ), (S. ’09)
2. k is large enough depending on g . (S. ’09)
3. Kg > 0 and k is small enough depending on g . (Rosenberg &S. ’11)
Results
I (*1) is true for (S2, g), if Kg ≥ 0 and k > 0, i.e. there are twoclosed k-magnetic geodesics in this case. (S. ’10)
I There is a closed k-magnetic geodesic, if χ(M) < 0, Kg ≥ −1and k > 1. (S. ’10)
I (*2) holds for (S2, g), i.e. there are two closed embeddedk-magnetic geodesics, under each of the following threeassumptions:
1. k > 0 and g is 14 -pinched (sup Kg < 4 inf Kg ), (S. ’09)
2. k is large enough depending on g . (S. ’09)
3. Kg > 0 and k is small enough depending on g . (Rosenberg &S. ’11)
Results
I (*1) is true for (S2, g), if Kg ≥ 0 and k > 0, i.e. there are twoclosed k-magnetic geodesics in this case. (S. ’10)
I There is a closed k-magnetic geodesic, if χ(M) < 0, Kg ≥ −1and k > 1. (S. ’10)
I (*2) holds for (S2, g), i.e. there are two closed embeddedk-magnetic geodesics, under each of the following threeassumptions:
1. k > 0 and g is 14 -pinched (sup Kg < 4 inf Kg ), (S. ’09)
2. k is large enough depending on g . (S. ’09)
3. Kg > 0 and k is small enough depending on g . (Rosenberg &S. ’11)
Results
I (*1) is true for (S2, g), if Kg ≥ 0 and k > 0, i.e. there are twoclosed k-magnetic geodesics in this case. (S. ’10)
I There is a closed k-magnetic geodesic, if χ(M) < 0, Kg ≥ −1and k > 1. (S. ’10)
I (*2) holds for (S2, g), i.e. there are two closed embeddedk-magnetic geodesics, under each of the following threeassumptions:
1. k > 0 and g is 14 -pinched (sup Kg < 4 inf Kg ), (S. ’09)
2. k is large enough depending on g . (S. ’09)
3. Kg > 0 and k is small enough depending on g . (Rosenberg &S. ’11)
Hopf tori in S3
Following Barros, Ferrandez, Lucas ’99 we consider
S3 = (z1, z2) : z1, z2 ∈ C, |z1|2 + |z2|2 = 4
and the Hopf map H : S3 → S2 defined by
H(z1, z2) :=1
4
(|z1|2 − |z2|2, 2z1z2
)∈ ∂B1(0) ⊂ R3.
For any metric metric g on S2, we define a metric g on S3 by
g(V ,W ) := H∗g(V ,W ) + θ(V )θ(W ),
where θ|x(V ) := 12〈ix ,V 〉R4 .
If γ is a closed (embedded) curve in S2, then H−1(γ) is a flat(embedded) torus in S3 with mean curvature
HgH−1(γ(t)) =
1
2kg (γ, t).
Hopf tori in S3
Consequently we find
I Given k : S2 → R positive, then there are two embedded toriin the round S3 with prescribed mean curvature k H.
I Given a 14 -pinched metric g on S2 and c > 0, then there are
two embedded tori in (S3, g) with constant mean curvature c .
An outline of the proof
Closed k-magnetic geodesics correspond to zeros of the vector fieldXg ,k on H2,2(S1,M) given by
Xk,g (γ) := (−D2t,g + 1)−1
(− Dt,g γ + |γ|gJg (γ)γ
).
Note that
TγH2,2(S1,M) = Periodic H2,2 − vector fields along γ.
For θ ∈ S1 = R/Z consider the action on H2,2(S1,M) defined by
(θ ∗ γ)(t) := γ(t + θ).
The vector field Xk,g is invariant under the S1-action and any zerocomes along with a S1 orbit of zeros.
An outline of the proof
I Step 1: Count zero orbits, i.e. define a S1-degree.
I Step 2: Compute the S1-degree in an unperturbed situation.e.g. constant curvature and k .
I Step 3: Prove compactness of the set of zero orbits and use ahomotopy argument.
An outline of the proof
I Step 1: Count zero orbits, i.e. define a S1-degree.
I Step 2: Compute the S1-degree in an unperturbed situation.e.g. constant curvature and k .
I Step 3: Prove compactness of the set of zero orbits and use ahomotopy argument.
An outline of the proof
I Step 1: Count zero orbits, i.e. define a S1-degree.
I Step 2: Compute the S1-degree in an unperturbed situation.e.g. constant curvature and k .
I Step 3: Prove compactness of the set of zero orbits and use ahomotopy argument.
Step 1: The S1-degree
We follow the degree theory of Tromba ’78 and give aS1-equivariant version. To this end we note:
I Fix a zero γ of Xk,g . By standard regularity results γ is inH4,2(S1,M) and the map θ 7→ θ ∗ γ is C 2 from S1 toH2,2(S1,M). Hence, γ is in the kernel of DgXk,g |γ .
I Define a vector field Wg on H2,2(S1,M) by
Wg (γ) = (−(Dt,g )2 + 1)−1γ.
The vector field Xk,g is orthogonal to Wg . Consequently,
DgXk,g |γ : TγH2,2(S1,M)→ 〈Wg (γ)〉⊥.
Note that γ /∈ 〈Wg (γ)〉⊥.
Step 1: The S1-degree
We follow the degree theory of Tromba ’78 and give aS1-equivariant version. To this end we note:
I Fix a zero γ of Xk,g . By standard regularity results γ is inH4,2(S1,M) and the map θ 7→ θ ∗ γ is C 2 from S1 toH2,2(S1,M). Hence, γ is in the kernel of DgXk,g |γ .
I Define a vector field Wg on H2,2(S1,M) by
Wg (γ) = (−(Dt,g )2 + 1)−1γ.
The vector field Xk,g is orthogonal to Wg . Consequently,
DgXk,g |γ : TγH2,2(S1,M)→ 〈Wg (γ)〉⊥.
Note that γ /∈ 〈Wg (γ)〉⊥.
Step 1: The S1-degree
DefinitionThe zero orbit S1 ∗ γ is called nondegenerate, if
DgXk,g |γ : 〈Wg (γ)〉⊥ −→ 〈Wg (γ)〉⊥
is an isomorphism.
DefinitionWe define the local S1-degree of a nondegenerate zero orbit S1 ∗ γby
degloc,S1(Xk,g ,S1 ∗ γ) := sgnDgXk,g |γ ,
where sgnDgXk,g |γ denotes the usual Leray-Schauder degree.Note that the map DgXk,g |γ is of the form identity − compact.
Step 1: The S1-degree
Let M be an open S1-invariant subset of curves in H2,2(S1,M).We assume that Xk,g is proper in M, i.e.
γ ∈M : Xk,g (γ) = 0
is compact.Using an equivariant version of the Sard-Smale lemma, theS1-degree χS1(Xk,g ,M) ∈ Z is defined by
χS1(Xk,g ,M) :=∑
S1∗γ⊂M whereYk,g (S1∗γ)=0
degloc,S1(Yk,g , S1 ∗ γ),
where Yk,g is a small perturbation of Xk,g with only finitely manycritical orbits in M, that are all nondegenerate.
The Poincare map
Let S1 ∗ γ be an isolated zero orbit of Xk,g and consider thecorresponding periodic orbit in the unit tangent bundle
Σ1M := (x ,V ) ∈ TM : |V |g = 1.
We fix a transversal section E in Σ1M at the pointθ := (γ(0), |γ(0)|−1γ(0)) and denote by P : B1 ∩ E → B2 ∩ E thePoincare map, where B1, B2 are open neighborhoods of θ.
Lemma (S. ’09, Nikishin ’74, Simon ’74)
Under the above assumptions θ is an isolated fixed point of P andthere holds
degloc,S1(Xk,g , S1 ∗ γ) = −i(P, θ) ≥ −1,
where i(P, θ) denotes the index of the isolated fixed point θ.
Step 2: Computation of the S1 degree
Fix (M, g0) with a constant curvature metric g0 andk0 > 0 , if M = S2
k0 >> 1, if M 6= S2,
and consider the set of curves
MA := γ ∈ H2,2(S1,M) : γ 6= 0, γ is Alexandrov embedded.
and additionally for M = S2
ME := γ ∈ H2,2(S1, S2) : γ 6= 0, γ is embedded.
Then the set of zero orbits in MA as well as in ME is parametrizedby M.
Step 2: Computation of the S1 degree
To compute the S1-degree, choose a Morse function k1 on M andreplace k0 by k0 + εk1.As ε→ 0+ we find
I To every critical point w ∈ M of k1 corresponds exactly onezero orbit S1 ∗ γw of Xk0+εk1,g0 .
I degloc,S1(Xk0+εk1,g0 ,S1 ∗ γw ) = − degloc,S1(∇k1,w).
I These are all zero orbits in MA or ME .
Hence
χS1(Xk0,g0 ,ME ) = χS1(Xk0,g0 ,MA)
= χS1(Xk0+εk1,g0 ,MA) = −χ(M).
Step 2: Computation of the S1 degree
To compute the S1-degree, choose a Morse function k1 on M andreplace k0 by k0 + εk1.As ε→ 0+ we find
I To every critical point w ∈ M of k1 corresponds exactly onezero orbit S1 ∗ γw of Xk0+εk1,g0 .
I degloc,S1(Xk0+εk1,g0 , S1 ∗ γw ) = − degloc,S1(∇k1,w).
I These are all zero orbits in MA or ME .
Hence
χS1(Xk0,g0 ,ME ) = χS1(Xk0,g0 ,MA)
= χS1(Xk0+εk1,g0 ,MA) = −χ(M).
Step 2: Computation of the S1 degree
To compute the S1-degree, choose a Morse function k1 on M andreplace k0 by k0 + εk1.As ε→ 0+ we find
I To every critical point w ∈ M of k1 corresponds exactly onezero orbit S1 ∗ γw of Xk0+εk1,g0 .
I degloc,S1(Xk0+εk1,g0 , S1 ∗ γw ) = − degloc,S1(∇k1,w).
I These are all zero orbits in MA or ME .
Hence
χS1(Xk0,g0 ,ME ) = χS1(Xk0,g0 ,MA)
= χS1(Xk0+εk1,g0 ,MA) = −χ(M).
Step 2: Computation of the S1 degree
To compute the S1-degree, choose a Morse function k1 on M andreplace k0 by k0 + εk1.As ε→ 0+ we find
I To every critical point w ∈ M of k1 corresponds exactly onezero orbit S1 ∗ γw of Xk0+εk1,g0 .
I degloc,S1(Xk0+εk1,g0 , S1 ∗ γw ) = − degloc,S1(∇k1,w).
I These are all zero orbits in MA or ME .
Hence
χS1(Xk0,g0 ,ME ) = χS1(Xk0,g0 ,MA)
= χS1(Xk0+εk1,g0 ,MA) = −χ(M).
Step 3: Compactness
A priori estimates: Fix (M, g) and γ ∈MA, i.e. there is animmersion F : B → M such that γ = F (∂B).Gauss-Bonnet applied to (B,F ∗g) yields
2π =
∫B
KF∗gdVF∗g +
∫∂B
kF∗gdSF∗g
≥ min(Kg )vol(B) +
∫γ
kgdSg
≥ min(Kg )vol(B) + L(γ) min(k)
I If Kg ≥ 0, then L(γ) ≤ 2π(min(k))−1.
I If Kg ≡ −1, then the (hyperbolic) isoperimetric inequalitygives L(γ) ≥ vol(B). Hence L(γ) ≤ 2π(min(k)− 1)−1.For the general case Kg ≥ −1 we use a conformal change ofthe metric in B.
Step 3: Compactness
A priori estimates: Fix (M, g) and γ ∈MA, i.e. there is animmersion F : B → M such that γ = F (∂B).Gauss-Bonnet applied to (B,F ∗g) yields
2π =
∫B
KF∗gdVF∗g +
∫∂B
kF∗gdSF∗g
≥ min(Kg )vol(B) +
∫γ
kgdSg
≥ min(Kg )vol(B) + L(γ) min(k)
I If Kg ≥ 0, then L(γ) ≤ 2π(min(k))−1.
I If Kg ≡ −1, then the (hyperbolic) isoperimetric inequalitygives L(γ) ≥ vol(B). Hence L(γ) ≤ 2π(min(k)− 1)−1.For the general case Kg ≥ −1 we use a conformal change ofthe metric in B.
Step 3: Compactness
A priori estimates: Fix (M, g) and γ ∈MA, i.e. there is animmersion F : B → M such that γ = F (∂B).Gauss-Bonnet applied to (B,F ∗g) yields
2π =
∫B
KF∗gdVF∗g +
∫∂B
kF∗gdSF∗g
≥ min(Kg )vol(B) +
∫γ
kgdSg
≥ min(Kg )vol(B) + L(γ) min(k)
I If Kg ≥ 0, then L(γ) ≤ 2π(min(k))−1.
I If Kg ≡ −1, then the (hyperbolic) isoperimetric inequalitygives L(γ) ≥ vol(B). Hence L(γ) ≤ 2π(min(k)− 1)−1.For the general case Kg ≥ −1 we use a conformal change ofthe metric in B.
Step 3: Compactness
This yields compactness in MA, because the limit of Alexandrovembedded locally convex curves remains Alexandrov embedded.Using the homotopy invariance of the S1-degree, we find
−χ(M) = χS1(Xk0,g0 ,MA) = χS1(Xk,g ,MA).
In particular, if M = S2, the S1-degree is −2.Since the local S1-degree of an isolated zero orbit is greater thanor equal to −1, there are at least two zero orbits.
Step 3: Compactness
We consider M = S2 and embedded in curves in ME .
I To obtain a contradiction, assume there are embedded curvesconverging to γ0, which has positive geodesic curvature and isnot embedded.
I Let (Ω0, g) be the interior of γ0 considered as a Riemanniansurface with boundary. Fix a touching point γ0(s1) = γ0(s2).The point γ0(s1) = γ0(s2) corresponds to two differentboundary points of Ω0.
I Let β be the curve of minimal length in Ω0 connecting thetwo boundary points.
I By the maximum principle β cannot touch the boundary of Ω0
and is therefore a geodesic in the interior of Ω0.
I Thus, β is a stable geodesic loop in S2.
Step 3: Compactness
We consider M = S2 and embedded in curves in ME .
I To obtain a contradiction, assume there are embedded curvesconverging to γ0, which has positive geodesic curvature and isnot embedded.
I Let (Ω0, g) be the interior of γ0 considered as a Riemanniansurface with boundary. Fix a touching point γ0(s1) = γ0(s2).The point γ0(s1) = γ0(s2) corresponds to two differentboundary points of Ω0.
I Let β be the curve of minimal length in Ω0 connecting thetwo boundary points.
I By the maximum principle β cannot touch the boundary of Ω0
and is therefore a geodesic in the interior of Ω0.
I Thus, β is a stable geodesic loop in S2.
Step 3: Compactness
We consider M = S2 and embedded in curves in ME .
I To obtain a contradiction, assume there are embedded curvesconverging to γ0, which has positive geodesic curvature and isnot embedded.
I Let (Ω0, g) be the interior of γ0 considered as a Riemanniansurface with boundary. Fix a touching point γ0(s1) = γ0(s2).The point γ0(s1) = γ0(s2) corresponds to two differentboundary points of Ω0.
I Let β be the curve of minimal length in Ω0 connecting thetwo boundary points.
I By the maximum principle β cannot touch the boundary of Ω0
and is therefore a geodesic in the interior of Ω0.
I Thus, β is a stable geodesic loop in S2.
Step 3: Compactness
We consider M = S2 and embedded in curves in ME .
I To obtain a contradiction, assume there are embedded curvesconverging to γ0, which has positive geodesic curvature and isnot embedded.
I Let (Ω0, g) be the interior of γ0 considered as a Riemanniansurface with boundary. Fix a touching point γ0(s1) = γ0(s2).The point γ0(s1) = γ0(s2) corresponds to two differentboundary points of Ω0.
I Let β be the curve of minimal length in Ω0 connecting thetwo boundary points.
I By the maximum principle β cannot touch the boundary of Ω0
and is therefore a geodesic in the interior of Ω0.
I Thus, β is a stable geodesic loop in S2.
Step 3: Compactness
We consider M = S2 and embedded in curves in ME .
I To obtain a contradiction, assume there are embedded curvesconverging to γ0, which has positive geodesic curvature and isnot embedded.
I Let (Ω0, g) be the interior of γ0 considered as a Riemanniansurface with boundary. Fix a touching point γ0(s1) = γ0(s2).The point γ0(s1) = γ0(s2) corresponds to two differentboundary points of Ω0.
I Let β be the curve of minimal length in Ω0 connecting thetwo boundary points.
I By the maximum principle β cannot touch the boundary of Ω0
and is therefore a geodesic in the interior of Ω0.
I Thus, β is a stable geodesic loop in S2.
Step 3: Compactness
Consequently, if ME is not compact, then there is a stablegeodesic loop β inside Ω0 ⊂ S2.Hence
I inj(g) ≤ 12L(β) < 1
4L(γ0), a contradiction if k >> 1.
I If Kg > 0 then from Klingenberg ’78 and Bonnet-Meyer
π(
sup Kg
)− 12 ≤ inj(g) ≤ 1
2L(β) ≤ 1
2π(
inf Kg
)− 12 ,
a contradiction, if g is 1/4-pinched.
Step 3: Compactness
Consequently, if ME is not compact, then there is a stablegeodesic loop β inside Ω0 ⊂ S2.Hence
I inj(g) ≤ 12L(β) < 1
4L(γ0), a contradiction if k >> 1.
I If Kg > 0 then from Klingenberg ’78 and Bonnet-Meyer
π(
sup Kg
)− 12 ≤ inj(g) ≤ 1
2L(β) ≤ 1
2π(
inf Kg
)− 12 ,
a contradiction, if g is 1/4-pinched.
The Legendrian Seifert conjecture
Closed magnetic geodesics correspond to periodic trajectories of avector field Φ on the unit tangent bundle of (S2, g)
Σ1S2 := (x ,V ) ∈ TS2 : |V |g = 1 ∼= RP3 1:2←−− S3
Lift Φ to a vector field Φ on S3.
The Legendrian Seifert conjecture
I Seifert 1950: Exists a vector field on S3 without periodic orbit?
I Kuperberg 1994: There is a smooth vector field on S3 withoutperiodic orbit.
I Hofer 1993: Every Reeb vector field V on (S3, λ) has aperiodic orbit.contact structure: λ ∧ dλ 6= 0, Reeb vector field: dλ(V , ·) = 0and λ(V ) = 1. (Weinstein conjecture, Taubes 2007.)
I Φ is Legendrian for the (standard) contact structure λ liftedfrom Σ1S2, i.e. λ(Φ) = 0.
I Open question: Existence of periodic orbits of Legendrianvector fields on S3 (with standard contact structure)?
The Legendrian Seifert conjecture
I Seifert 1950: Exists a vector field on S3 without periodic orbit?
I Kuperberg 1994: There is a smooth vector field on S3 withoutperiodic orbit.
I Hofer 1993: Every Reeb vector field V on (S3, λ) has aperiodic orbit.contact structure: λ ∧ dλ 6= 0, Reeb vector field: dλ(V , ·) = 0and λ(V ) = 1. (Weinstein conjecture, Taubes 2007.)
I Φ is Legendrian for the (standard) contact structure λ liftedfrom Σ1S2, i.e. λ(Φ) = 0.
I Open question: Existence of periodic orbits of Legendrianvector fields on S3 (with standard contact structure)?
The Legendrian Seifert conjecture
I Seifert 1950: Exists a vector field on S3 without periodic orbit?
I Kuperberg 1994: There is a smooth vector field on S3 withoutperiodic orbit.
I Hofer 1993: Every Reeb vector field V on (S3, λ) has aperiodic orbit.contact structure: λ ∧ dλ 6= 0, Reeb vector field: dλ(V , ·) = 0and λ(V ) = 1. (Weinstein conjecture, Taubes 2007.)
I Φ is Legendrian for the (standard) contact structure λ liftedfrom Σ1S2, i.e. λ(Φ) = 0.
I Open question: Existence of periodic orbits of Legendrianvector fields on S3 (with standard contact structure)?