Closed magnetic geodesics on Riemann surfaces

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Transcript of Closed magnetic geodesics on Riemann surfaces

Closed magnetic geodesics on Riemann surfacesMatthias Schneider
April 8. 2011, Sevilla
The problem Given
I an oriented (compact) surface (M, g) with a Riemannian metric g ,
I a smooth (positive) function k : M → R.
Problem: Existence of a closed immersed curve γ : S1 → M with geodesic 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 orientation and metric.
I These curves are called magnetic geodesics. They correspond to trajectories of a charged particle on M in a magnetic field with magnetic form kdµg and solve
Dt,g γ = |γ|gk(γ)Jg (γ)γ.
The problem Given
I an oriented (compact) surface (M, g) with a Riemannian metric g ,
I a smooth (positive) function k : M → R.
Problem: Existence of a closed immersed curve γ : S1 → M with geodesic 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 orientation and metric.
I These curves are called magnetic geodesics. They correspond to trajectories of a charged particle on M in a magnetic field with magnetic form kdµg and solve
Dt,g γ = |γ|gk(γ)Jg (γ)γ.
The problem Given
I an oriented (compact) surface (M, g) with a Riemannian metric g ,
I a smooth (positive) function k : M → R.
Problem: Existence of a closed immersed curve γ : S1 → M with geodesic 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 orientation and metric.
I These curves are called magnetic geodesics. They correspond to trajectories of a charged particle on M in a magnetic 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 Magnetic geodesics with 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 Magnetic geodesics with 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 Magnetic geodesics with 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 Magnetic geodesics with k0 = 1.
The hyperbolic geometry
Example 2: The hyperbolic plane (H, gH) with k ≡ k0.
I H = B1(0) ⊂ R2
I Gauss curvature KgH ≡ −1
I Magnetic geodesics for k0 = 2
The hyperbolic geometry
Example 2: The hyperbolic plane (H, gH) with k ≡ k0.
I H = B1(0) ⊂ R2
I Gauss curvature KgH ≡ −1
I Magnetic geodesics for k0 = 2
The hyperbolic geometry
Example 2: The hyperbolic plane (H, gH) with k ≡ k0.
I H = B1(0) ⊂ R2
I Gauss curvature KgH ≡ −1
I Magnetic geodesics for k0 = 2
The hyperbolic geometry
Example 2: The hyperbolic plane (H, gH) with k ≡ k0.
I H = B1(0) ⊂ R2
I Gauss curvature KgH ≡ −1
I Magnetic geodesics for k0 = 2
The hyperbolic geometry
Example 2: The hyperbolic plane (H, gH) with k ≡ k0.
I H = B1(0) ⊂ R2
I Gauss curvature KgH ≡ −1
I Magnetic geodesics for k0 = 2
The hyperbolic geometry
Example 2: The hyperbolic plane (H, gH) with k ≡ k0.
I H = B1(0) ⊂ R2
I Gauss curvature KgH ≡ −1
I Magnetic geodesics for k0 = 2
Methods and approaches
There is a vast literature on the existence of closed magnetic geodesics.
I Theory of dynamical systems: Arnold ’86, Ginzburg ’96 Magnetic geodesics are periodic orbits of a twisted Hamiltonian flow with H = 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 ’92 Minimize E (γ) := 1
2
If dθ = kµg , then ” ∫ B kµg” =
∫ γ θ.
I Aubry-Mather-Theorie: Contreras, Macarini, Paternain ’04 Mane’s critical value, existence on compact surfaces, if kµg is exact.
Methods and approaches
There is a vast literature on the existence of closed magnetic geodesics.
I Theory of dynamical systems: Arnold ’86, Ginzburg ’96 Magnetic geodesics are periodic orbits of a twisted Hamiltonian flow with H = 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 ’92 Minimize E (γ) := 1
2
If dθ = kµg , then ” ∫ B kµg” =
∫ γ θ.
I Aubry-Mather-Theorie: Contreras, Macarini, Paternain ’04 Mane’s critical value, existence on compact surfaces, if kµg is exact.
Methods and approaches
There is a vast literature on the existence of closed magnetic geodesics.
I Theory of dynamical systems: Arnold ’86, Ginzburg ’96 Magnetic geodesics are periodic orbits of a twisted Hamiltonian flow with H = 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 ’92 Minimize E (γ) := 1
2
If dθ = kµg , then ” ∫ B kµg” =
∫ γ θ.
I Aubry-Mather-Theorie: Contreras, Macarini, Paternain ’04 Mane’s critical value, existence on compact surfaces, if kµg is exact.
Methods and approaches
There is a vast literature on the existence of closed magnetic geodesics.
I Theory of dynamical systems: Arnold ’86, Ginzburg ’96 Magnetic geodesics are periodic orbits of a twisted Hamiltonian flow with H = 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 ’92 Minimize E (γ) := 1
2
If dθ = kµg , then ” ∫ B kµg” =
∫ γ θ.
I Aubry-Mather-Theorie: Contreras, Macarini, Paternain ’04 Mane’s critical value, existence on compact surfaces, if kµg is exact.
Open problems
I Conjecture (*1): (Arnold ’81) If (M, g) is compact oriented surface and k is positive, then there is a closed k-magnetic geodesic. More precisely, there are at least two for M = S2 and at least three 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 ’10 On (S2, g) with k positive, there is an embedded closed k-magnetic geodesic.
Open problems
I Conjecture (*1): (Arnold ’81) If (M, g) is compact oriented surface and k is positive, then there is a closed k-magnetic geodesic. More precisely, there are at least two for M = S2 and at least three 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 ’10 On (S2, g) with k positive, there is an embedded closed k-magnetic geodesic.
Open problems
I Conjecture (*1): (Arnold ’81) If (M, g) is compact oriented surface and k is positive, then there is a closed k-magnetic geodesic. More precisely, there are at least two for M = S2 and at least three 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 ’10 On (S2, g) with k positive, there is an embedded closed k-magnetic geodesic.
Results
I (*1) is true for (S2, g), if Kg ≥ 0 and k > 0, i.e. there are two closed k-magnetic geodesics in this case. (S. ’10)
I There is a closed k-magnetic geodesic, if χ(M) < 0, Kg ≥ −1 and k > 1. (S. ’10)
I (*2) holds for (S2, g), i.e. there are two closed embedded k-magnetic geodesics, under each of the following three assumptions:
1. k > 0 and g is 1 4 -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 two closed k-magnetic geodesics in this case. (S. ’10)
I There is a closed k-magnetic geodesic, if χ(M) < 0, Kg ≥ −1 and k > 1. (S. ’10)
I (*2) holds for (S2, g), i.e. there are two closed embedded k-magnetic geodesics, under each of the following three assumptions:
1. k > 0 and g is 1 4 -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 two closed k-magnetic geodesics in this case. (S. ’10)
I There is a closed k-magnetic geodesic, if χ(M) < 0, Kg ≥ −1 and k > 1. (S. ’10)
I (*2) holds for (S2, g), i.e. there are two closed embedded k-magnetic geodesics, under each of the following three assumptions:
1. k > 0 and g is 1 4 -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 two closed k-magnetic geodesics in this case. (S. ’10)
I There is a closed k-magnetic geodesic, if χ(M) < 0, Kg ≥ −1 and k > 1. (S. ’10)
I (*2) holds for (S2, g), i.e. there are two closed embedded k-magnetic geodesics, under each of the following three assumptions:
1. k > 0 and g is 1 4 -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 two closed k-magnetic geodesics in this case. (S. ’10)
I There is a closed k-magnetic geodesic, if χ(M) < 0, Kg ≥ −1 and k > 1. (S. ’10)
I (*2) holds for (S2, g), i.e. there are two closed embedded k-magnetic geodesics, under each of the following three assumptions:
1. k > 0 and g is 1 4 -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 two closed k-magnetic geodesics in this case. (S. ’10)
I There is a closed k-magnetic geodesic, if χ(M) < 0, Kg ≥ −1 and k > 1. (S. ’10)
I (*2) holds for (S2, g), i.e. there are two closed embedded k-magnetic geodesics, under each of the following three assumptions:
1. k > 0 and g is 1 4 -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
) ∈ ∂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 ) := 1 2ix ,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)) =
Consequently we find
I Given k : S2 → R positive, then there are two embedded tori in the round S3 with prescribed mean curvature k H.
I Given a 1 4 -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 field Xg ,k on H2,2(S1,M) given by
Xk,g (γ) := (−D2 t,g + 1)−1
( − Dt,g γ + |γ|gJg (γ)γ
) .
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 zero comes 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 a homotopy 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 a homotopy 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 a homotopy argument.
Step 1: The S1-degree
We follow the degree theory of Tromba ’78 and give a S1-equivariant version. To this end we note:
I Fix a zero γ of Xk,g . By standard regularity results γ is in H4,2(S1,M) and the map θ 7→ θ ∗ γ is C 2 from S1 to H2,2(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 a S1-equivariant version. To this end we note:
I Fix a zero γ of Xk,g . By standard regularity results γ is in H4,2(S1,M) and the map θ 7→ θ ∗ γ is C 2 from S1 to H2,2(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
Definition The zero orbit S1 ∗ γ is called nondegenerate, if
DgXk,g |γ : Wg (γ)⊥ −→ Wg (γ)⊥
is an isomorphism.
Definition We define the local S1-degree of a nondegenerate zero orbit S1 ∗ γ by
degloc,S1(Xk,g ,S 1 ∗ 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, the S1-degree χS1(Xk,g ,M) ∈ Z is defined by
χS1(Xk,g ,M) := ∑
degloc,S1(Yk,g , S 1 ∗ γ),
where Yk,g is a small perturbation of Xk,g with only finitely many critical orbits in M, that are all nondegenerate.
The Poincare map
Let S1 ∗ γ be an isolated zero orbit of Xk,g and consider the corresponding periodic orbit in the unit tangent bundle
Σ1M := {(x ,V ) ∈ TM : |V |g = 1}.
We fix a transversal section E in Σ1M at the point θ := (γ(0), |γ(0)|−1γ(0)) and denote by P : B1 ∩ E → B2 ∩ E the Poincare 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 and there holds
degloc,S1(Xk,g , S 1 ∗ 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 and{ k0 > 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 parametrized by M.
Step 2: Computation of the S1 degree
To compute the S1-degree, choose a Morse function k1 on M and replace k0 by k0 + εk1. As ε→ 0+ we find
I To every critical point w ∈ M of k1 corresponds exactly one zero orbit S1 ∗ γw of Xk0+εk1,g0 .
I degloc,S1(Xk0+εk1,g0 ,S 1 ∗ γw ) = − degloc,S1(∇k1,w).
I These are all zero orbits in MA or ME .
Hence
Step 2: Computation of the S1 degree
To compute the S1-degree, choose a Morse function k1 on M and replace k0 by k0 + εk1. As ε→ 0+ we find
I To every critical point w ∈ M of k1 corresponds exactly one zero orbit S1 ∗ γw of Xk0+εk1,g0 .
I degloc,S1(Xk0+εk1,g0 , S 1 ∗ γw ) = − degloc,S1(∇k1,w).
I These are all zero orbits in MA or ME .
Hence
Step 2: Computation of the S1 degree
To compute the S1-degree, choose a Morse function k1 on M and replace k0 by k0 + εk1. As ε→ 0+ we find
I To every critical point w ∈ M of k1 corresponds exactly one zero orbit S1 ∗ γw of Xk0+εk1,g0 .
I degloc,S1(Xk0+εk1,g0 , S 1 ∗ γw ) = − degloc,S1(∇k1,w).
I…