Periodic orbits on non-compact hypersurfaces in cotangent bundles
Joint work with: J.B. van den Berg, F. Pasquotto, R.C.A.M. Vandervorst
Thomas Rot
VU University Amsterdam
31 July 2012
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Table of Contents
The Weinstein conjecture
The non-compact setting
Analysis (PS-condition)
Topology (candidate critical points)
Results
Outlook
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Introduction
Hamiltonian dynamics
(W , ω) a symplectic manifold, H : W → R Hamiltonian.
ω(XH , −) = −dH
defines the Hamiltonian vector field XH .
XH is tangent to Σ = H −1 (0).
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Introduction
Hamiltonian dynamics
(W , ω) a symplectic manifold, H : W → R Hamiltonian.
ω(XH , −) = −dH
defines the Hamiltonian vector field XH .
XH is tangent to Σ = H −1 (0).
Question
When does Σ contain a periodic orbit/closed characteristic?
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Introduction
The Weinstein conjecture
Weinstein Conjecture
If Σ is closed and of contact type, and H 1 (Σ) = 0, then Σ contains a
closed characteristic.
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Introduction
Results on the Weinstein conjecture
Hypersurfaces in symplectic manifolds
Rabinowitz (star-shaped compact hypersurfaces in R2n )
Viterbo (compact hypersurfaces of contact type in R2n )
Hofer and Viterbo (compact hypersurfaces of contact type in
cotangent bundles of simply connected manifolds)
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Introduction
Results on the Weinstein conjecture
Hypersurfaces in symplectic manifolds
Rabinowitz (star-shaped compact hypersurfaces in R2n )
Viterbo (compact hypersurfaces of contact type in R2n )
Hofer and Viterbo (compact hypersurfaces of contact type in
cotangent bundles of simply connected manifolds)
Contact manifolds
Hofer (S 3 with any contact structure)
Taubes (All closed contact 3-manifolds)
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Introduction
Results on the Weinstein conjecture
Hypersurfaces in symplectic manifolds
Rabinowitz (star-shaped compact hypersurfaces in R2n )
Viterbo (compact hypersurfaces of contact type in R2n )
Hofer and Viterbo (compact hypersurfaces of contact type in
cotangent bundles of simply connected manifolds)
Contact manifolds
Hofer (S 3 with any contact structure)
Taubes (All closed contact 3-manifolds)
All results require compactness in the geometry.
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Introduction
Results on the Weinstein conjecture
Hypersurfaces in symplectic manifolds
Rabinowitz (star-shaped compact hypersurfaces in R2n )
Viterbo (compact hypersurfaces of contact type in R2n )
Hofer and Viterbo (compact hypersurfaces of contact type in
cotangent bundles of simply connected manifolds)
Contact manifolds
Hofer (S 3 with any contact structure)
Taubes (All closed contact 3-manifolds)
All results require compactness in the geometry.
Van den Berg, Pasquotto, Vandervorst (non-compact mechanical
hypersurfaces in R2n )
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The Setting
Non-compact hypersurfaces
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The Setting
Non-compact hypersurfaces
M non-compact Riemannian manifold of compactly supported
geometry
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The Setting
Non-compact hypersurfaces
M non-compact Riemannian manifold of compactly supported
geometry
The Hamiltonian H : T ∗ M → R is
1
H(q, p) = |p|2 + V (q)
2
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The Setting
Non-compact hypersurfaces
M non-compact Riemannian manifold of compactly supported
geometry
The Hamiltonian H : T ∗ M → R is
1
H(q, p) = |p|2 + V (q)
2
A geometric condition on the potential V , such that Σ = H −1 (0) is
of uniform contact type.
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The Setting
The Action Functional
Closed orbits are critical points of the action functional
Z
e −τ 0
A(c, τ ) =
|c (s)|2 − e τ V (c(s))ds.
2
1
S
Defined on the free loop space ΛM × R.
The τ parameter keeps track of the period.
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Methods
Linking (Generalized Mountain-Pass Theorem)
A standard technique for finding critical points is linking
Identify linking sets A, B ⊂ ΛM × R. Non-trivial inclusion
Hk (A) → Hk (ΛM × R \ B)
for some k
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Methods
Linking (Generalized Mountain-Pass Theorem)
A standard technique for finding critical points is linking
Identify linking sets A, B ⊂ ΛM × R. Non-trivial inclusion
Hk (A) → Hk (ΛM × R \ B)
for some k
The functional satisfies bounds
A|A ≤ a,
and
A|B ≥ b,
with − ∞ < a < b < ∞.
A minimax gives critical values of A
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Analysis
The Palais-Smale condition
If A satisfies Palais-Smale (PS), critical values contain critical points
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Analysis
The Palais-Smale condition
If A satisfies Palais-Smale (PS), critical values contain critical points
The functional does A not satisfy PS on non-compact manifolds!
However a small perturbation does
Z −τ
e
A (c, τ ) =
|c 0 (s)|2 − e τ V (c(s))ds + (e −τ + e τ /2 )
2
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Analysis
The Palais-Smale condition
If A satisfies Palais-Smale (PS), critical values contain critical points
The functional does A not satisfy PS on non-compact manifolds!
However a small perturbation does
Z −τ
e
A (c, τ ) =
|c 0 (s)|2 − e τ V (c(s))ds + (e −τ + e τ /2 )
2
Penalizes short and long orbits
Penalized functional satisfies Palais-Smale
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Analysis
A nice property of A
Lemma
If (c , τ ) are critical points for A , with uniform bounds
0 < a1 < A (c , τ ) < a2 < ∞,
then there is a subsequence such that
(c, τ ) = lim (c , τ ),
→0
exists and is a critical point of the unpenalized functional A.
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Linking
Topology
Linking sets are needed.
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Linking
Topology
Linking sets are needed.
We assume that Hn+k (Σ) 6= 0 and Hk+1 (M) = 0 for some k ≥ 0
The topology of Σ is reflected in the projection N = π(Σ) ⊂ M
By the Gysin sequence there is a non-zero class in Hk (M − N) that is
zero in Hk (M).
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Linking
Topology
Linking sets are needed.
We assume that Hn+k (Σ) 6= 0 and Hk+1 (M) = 0 for some k ≥ 0
The topology of Σ is reflected in the projection N = π(Σ) ⊂ M
By the Gysin sequence there is a non-zero class in Hk (M − N) that is
zero in Hk (M).
Figure: Example
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Figure: Non-Example
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Linking
The Loop space
Embed M in ΛM × R, as constant loops, in the connected component
with contractible loops. We lift the link to ΛM × R.
By construction we have bounds 0 < a < b < ∞
A |A ≤ a
and A |B ≥ b.
The linking theorem gives critical points of A!
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Linking
The Loop space
Embed M in ΛM × R, as constant loops, in the connected component
with contractible loops. We lift the link to ΛM × R.
By construction we have bounds 0 < a < b < ∞
A |A ≤ a
and A |B ≥ b.
The linking theorem gives critical points of A!
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Results
Conditions at infinity
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Results
Conditions at infinity
Definition (Asymptotic Regularity)
Let (M, g ) be a Riemannian manifold. A potential V : M → R is
asymptotically regular if there exists a compact K ⊂ M and a constant
v > 0 such that
|gradV (q)| ≥ v
for q ∈ M \ K
and
||HessV (q)||
→ 0,
|gradV (q)|
as d(q, K ) → ∞.
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Results
Conditions at infinity
Definition (Asymptotic Regularity)
Let (M, g ) be a Riemannian manifold. A potential V : M → R is
asymptotically regular if there exists a compact K ⊂ M and a constant
v > 0 such that
|gradV (q)| ≥ v
for q ∈ M \ K
and
||HessV (q)||
→ 0,
|gradV (q)|
as d(q, K ) → ∞.
Definition (Compactly supported geometry)
A non-compact Riemannian manifold is of compactly supported geometry
if the curvature vanishes outside a compact set.
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Results
Results
Theorem
Let (M, g ) be a complete orientable Riemannian n-dimensional manifold of
compactly supported geometry, and Σ = H −1 (0) asymptotically regular.
Assume that there exists a 0 ≤ k ≤ n − 1 such that the following
topological conditions hold
Hk+n (Σ) 6= 0,
Hk+1 (ΛM) = 0,
Hk+2 (ΛM) = 0.
Then Σ contains a closed characteristic, which is contractible on M.
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Example
Example
Take M = S 5 × R ∼
= R6 \ pt with metric
ds 2 = f (r )dsS2 5 + dr 2
with f non-zero and equal to r 2 outside [−1, 1]
compactly supported geometry and complete. Potential
V (q) =
ρ(q) 2
q1 + q22 − q32 − q42 − q52 − q62 ) − C ,
2
with ρ a bump function.
H7 (Σ) = Z2
H2 (ΛM) = 0
H3 (ΛM) = 0
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Outlook
Outlook
The conditions on the homology of the loop space are too strict.
Other way of choosing linking sets?
Relax compactly supported geometry to bounded geometry
What about non-contractible loops?
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