On the course

(1)

Orthogonal geodesic chords on Riemannian manifolds with concave boundary and

applications

Fabio Giannoni, Paolo Piccione

Universit`a di Camerino (Italy), Universidade de S˜ao Paulo (Brazil)

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On the course

Our goal: Prove the existence of multiple orthogonal geodesic chords in a class of compact Riemannian manifolds with boundary

School in Nonlinear Analysis and Calculus of Variations – p. 2/68

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On the course

Our goal: Prove the existence of multiple orthogonal geodesic chords in a class of compact Riemannian manifolds with boundary

Method: develop a non smooth Ljusternik–Schnirelmann

theory

(4)

On the course

Our goal: Prove the existence of multiple orthogonal geodesic chords in a class of compact Riemannian manifolds with boundary

Method: develop a non smooth Ljusternik–Schnirelmann theory

Applications: Prove the existence of:

(5)

On the course

Our goal: Prove the existence of multiple orthogonal geodesic chords in a class of compact Riemannian manifolds with boundary

Method: develop a non smooth Ljusternik–Schnirelmann theory

Applications: Prove the existence of:

multiple brake orbits for a class of Hamiltonian

problems

(6)

On the course

Our goal: Prove the existence of multiple orthogonal geodesic chords in a class of compact Riemannian manifolds with boundary

Method: develop a non smooth Ljusternik–Schnirelmann theory

Applications: Prove the existence of:

multiple brake orbits for a class of Hamiltonian problems

multiple homoclinic orbits for a class of Lagrangian systems

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Riemannian geometry

M smooth manifold

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Riemannian geometry

M smooth manifold

g symmetric, positive definite (2, 0) -tensor on M

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Riemannian geometry

M smooth manifold

g symmetric, positive definite (2, 0) -tensor on M

∇ Levi–Civita connection of g

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Riemannian geometry

M smooth manifold

g symmetric, positive definite (2, 0) -tensor on M

∇ Levi–Civita connection of g Σ ⊂ M hypersurface

S

_n

: T

_x

Σ × T

_x

Σ → R second fundamental form of Σ

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Riemannian geometry

M smooth manifold

g symmetric, positive definite (2, 0) -tensor on M

∇ Levi–Civita connection of g Σ ⊂ M hypersurface

S

_n

: T

_x

Σ × T

_x

Σ → R second fundamental form of Σ S

_n

(v, w) = g ∇

_v

W, n

_x

symmetric bilinear form

W extension of w , n

_x

normal vector to Σ at x .

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Riemannian geometry

M smooth manifold

g symmetric, positive definite (2, 0) -tensor on M

∇ Levi–Civita connection of g Σ ⊂ M hypersurface

S

_n

: T

_x

Σ × T

_x

Σ → R second fundamental form of Σ S

_n

(v, w) = g ∇

_v

W, n

_x

symmetric bilinear form

W extension of w , n

_x

normal vector to Σ at x .

Obs.: S

_n

is the Hessian of the map p 7→ dist

^∗

(p, Σ) .

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Riemannian geometry

M smooth manifold

g symmetric, positive definite (2, 0) -tensor on M

∇ Levi–Civita connection of g Σ ⊂ M hypersurface

S

_n

: T

_x

Σ × T

_x

Σ → R second fundamental form of Σ S

_n

(v, w) = g ∇

_v

W, n

_x

symmetric bilinear form

W extension of w , n

_x

normal vector to Σ at x .

“signed distance”

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Convex and Concave domains

(M, g ) Riemannian manifold Ω ⊂ M open subset, Ω = Ω S

∂ Ω

Ω

^γ

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Convex and Concave domains

(M, g ) Riemannian manifold Ω ⊂ M open subset, Ω = Ω S

∂ Ω

Definition. Ω is said to be convex if for all geodesic γ : [a, b] → Ω with γ (a), γ (b) ∈ Ω , then γ [a, b]

⊂ Ω . Ω is concave if M \ Ω is convex.

Ω

^γ

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Convex and Concave domains

(M, g ) Riemannian manifold Ω ⊂ M open subset, Ω = Ω S

∂ Ω

Definition. Ω is said to be convex if for all geodesic γ : [a, b] → Ω with γ (a), γ (b) ∈ Ω , then γ [a, b]

⊂ Ω . Ω is concave if M \ Ω is convex.

Ω is strongly concave if S

_n

is positive definite, where n is inward pointing.

Ω

^γ

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Convex and Concave domains

(M, g ) Riemannian manifold Ω ⊂ M open subset, Ω = Ω S

∂ Ω

Definition. Ω is said to be convex if for all geodesic γ : [a, b] → Ω with γ (a), γ (b) ∈ Ω , then γ [a, b]

⊂ Ω . Ω is concave if M \ Ω is convex.

Ω is strongly concave if S

_n

is positive definite, where n is inward pointing.

C

²

-open condition

Ω

^γ

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Convex and Concave domains

(M, g ) Riemannian manifold Ω ⊂ M open subset, Ω = Ω S

∂ Ω

Definition. Ω is said to be convex if for all geodesic γ : [a, b] → Ω with γ (a), γ (b) ∈ Ω , then γ [a, b]

⊂ Ω . Ω is concave if M \ Ω is convex.

Ω is strongly concave if S

_n

is positive definite, where n is inward pointing.

Lemma. Ω strongly concave = ⇒ Ω concave.

Ω

^γ

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Convex and Concave domains

(M, g ) Riemannian manifold Ω ⊂ M open subset, Ω = Ω S

∂ Ω

Definition. Ω is said to be convex if for all geodesic γ : [a, b] → Ω with γ (a), γ (b) ∈ Ω , then γ [a, b]

⊂ Ω . Ω is concave if M \ Ω is convex.

Ω is strongly concave if S

_n

is positive definite, where n is inward pointing.

γ

geodesics starting

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The boundary of Ω

Assume ∂ Ω is smooth

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The boundary of Ω

Assume ∂ Ω is smooth

∃ a smooth map φ : M → R with:

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The boundary of Ω

Assume ∂ Ω is smooth

∃ a smooth map φ : M → R with:

Ω = φ

⁻¹

]−∞, 0[

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The boundary of Ω

Assume ∂ Ω is smooth

∃ a smooth map φ : M → R with:

Ω = φ

⁻¹

]−∞, 0[

∂ Ω = φ

⁻¹

(0)

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The boundary of Ω

Assume ∂ Ω is smooth

∃ a smooth map φ : M → R with:

Ω = φ

⁻¹

]−∞, 0[

∂ Ω = φ

⁻¹

(0) dφ 6= 0 on ∂ Ω

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The boundary of Ω

Assume ∂ Ω is smooth

∃ a smooth map φ : M → R with:

Ω = φ

⁻¹

]−∞, 0[

∂ Ω = φ

⁻¹

(0) dφ 6= 0 on ∂ Ω

|φ(q)| = dist q, ∂ Ω

for q near ∂ Ω .

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The boundary of Ω

Assume ∂ Ω is smooth

∃ a smooth map φ : M → R with:

Ω = φ

⁻¹

]−∞, 0[

∂ Ω = φ

⁻¹

(0) dφ 6= 0 on ∂ Ω

|φ(q)| = dist q, ∂ Ω

for q near ∂ Ω . Observe: Hess(φ) = −S

_∇_φ

on T ∂ Ω

.

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The boundary of Ω

Assume ∂ Ω is smooth

∃ a smooth map φ : M → R with:

Ω = φ

⁻¹

]−∞, 0[

∂ Ω = φ

⁻¹

(0) dφ 6= 0 on ∂ Ω

|φ(q)| = dist q, ∂ Ω

for q near ∂ Ω . Observe: Hess(φ) = −S

_∇_φ

on T ∂ Ω

.

Back to proof

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Orthogonal geodesic chords

Def.: An orthogonal geodesic chord (OGC) in Ω is a non constant geodesic γ : [a, b] → Ω with γ (a), γ (b) ∈ ∂ Ω and

˙

γ (a), γ ˙ (b) ∈ T ∂ Ω

_⊥

.

Ω

γ

Ω

γ

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Orthogonal geodesic chords

Def.: An orthogonal geodesic chord (OGC) in Ω is a non constant geodesic γ : [a, b] → Ω with γ (a), γ (b) ∈ ∂ Ω and

˙

γ (a), γ ˙ (b) ∈ T ∂ Ω

_⊥

.

Ω

γ

Ω

γ

A weak orthogonal geodesic chord

(WOCG).

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Some examples – 1

Ω

Ω ∼ = annulus: S

^m⁻¹

× [0, 1]

An OGC is crossing if its endpoints are in distinct con- nected components of ∂Ω. It is easy to prove the existence of one crossing OGC whose length equals the distance between the two connected components of

∂Ω.

g

W

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Some examples – 1

Ω

Ω ∼ = annulus: S

^m⁻¹

× [0, 1]

An OGC is crossing if its endpoints are in distinct con- nected components of ∂Ω. It is easy to prove the existence of one crossing OGC whose length equals the distance between the two connected components of

∂Ω.

There may be only one OGC:

_g

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Some examples – 2

If Ω is convex, then it is proven the existence of at least two crossing OGC’s (Giannoni-Majer, DGA 1997).

e D

g

g₂

1 W_e

1

D₂

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Some examples – 2

If Ω is convex, then it is proven the existence of at least two crossing OGC’s (Giannoni-Majer, DGA 1997).

This is an optimal result

(in all dimensions):

Euclidean metric

g (x) = ψ |x|

· g

₀

(x), ψ : R

⁺

→ R

⁺

convexity of the annulus

1 2 ψ

^′

(1) + ψ(1)≥0, ψ

^′

(2) + ψ(2)≤0.

Ω

_ε

=

x ∈ R

^m

: 1 < |x|, |x − ε| < 2 .

e D

g

g₂

1 W_e

1

D₂

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Getting rid of WOGC’s

Proposition: Assume:

1. ∂ Ω compact and Ω strongly concave.

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Getting rid of WOGC’s

Proposition: Assume:

1. ∂ Ω compact and Ω strongly concave.

2. ∃ only a finite number of (crossing) OGC’s in Ω.

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Getting rid of WOGC’s

Proposition: Assume:

1. ∂ Ω compact and Ω strongly concave.

2. ∃ only a finite number of (crossing) OGC’s in Ω.

Then ∃ Ω

^′

⊂ Ω open with:

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Getting rid of WOGC’s

Proposition: Assume:

1. ∂ Ω compact and Ω strongly concave.

2. ∃ only a finite number of (crossing) OGC’s in Ω.

Then ∃ Ω

^′

⊂ Ω open with:

Ω

^′

diffeomorphic to Ω , ∂ Ω

^′

smooth;

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Getting rid of WOGC’s

Proposition: Assume:

1. ∂ Ω compact and Ω strongly concave.

2. ∃ only a finite number of (crossing) OGC’s in Ω.

Then ∃ Ω

^′

⊂ Ω open with:

Ω

^′

diffeomorphic to Ω , ∂ Ω

^′

smooth;

Ω

^′

strongly concave;

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Getting rid of WOGC’s

Proposition: Assume:

1. ∂ Ω compact and Ω strongly concave.

2. ∃ only a finite number of (crossing) OGC’s in Ω.

Then ∃ Ω

^′

⊂ Ω open with:

Ω

^′

diffeomorphic to Ω , ∂ Ω

^′

smooth;

Ω

^′

strongly concave;

the number of (crossing) OGC’s in Ω

^′

is ≤ number of

(crossing) OGC’s in Ω ;

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Getting rid of WOGC’s

Proposition: Assume:

1. ∂ Ω compact and Ω strongly concave.

2. ∃ only a finite number of (crossing) OGC’s in Ω.

Then ∃ Ω

^′

⊂ Ω open with:

Ω

^′

diffeomorphic to Ω , ∂ Ω

^′

smooth;

Ω

^′

strongly concave;

the number of (crossing) OGC’s in Ω

^′

is ≤ number of (crossing) OGC’s in Ω ;

there is no WOGC in Ω

^′

.

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Getting rid of WOGC’s

Proposition: Assume:

1. ∂ Ω compact and Ω strongly concave.

2. ∃ only a finite number of (crossing) OGC’s in Ω.

Then ∃ Ω

^′

⊂ Ω open with:

Ω

^′

diffeomorphic to Ω , ∂ Ω

^′

smooth;

Ω

^′

strongly concave;

the number of (crossing) OGC’s in Ω

^′

is ≤ number of

(crossing) OGC’s in Ω ;

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Proof

Ω

^′

= φ

⁻¹

]−∞, −δ[

, with δ > 0 small.

^(recall φ)

Observe:

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Proof

Ω

^′

= φ

⁻¹

]−∞, −δ[

, with δ > 0 small.

^(recall φ)

Observe:

by continuity, dφ 6= 0 in φ

⁻¹

[−δ, 0]

, so ∂ Ω

^′

is smooth;

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Proof

Ω

^′

= φ

⁻¹

]−∞, −δ[

, with δ > 0 small.

^(recall φ)

Observe:

by continuity, dφ 6= 0 in φ

⁻¹

[−δ, 0]

, so ∂ Ω

^′

is smooth;

Ω

^′

is strongly concave, by continuity of Hess(φ) and compactness of ∂ Ω ;

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Proof

Ω

^′

= φ

⁻¹

]−∞, −δ[

, with δ > 0 small.

^(recall φ)

Observe:

by continuity, dφ 6= 0 in φ

⁻¹

[−δ, 0]

, so ∂ Ω

^′

is smooth;

Ω

^′

is strongly concave, by continuity of Hess(φ) and compactness of ∂ Ω ;

if δ < foc(∂ Ω) , every OGC in Ω

^′

can be extended to an

OGC in Ω

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Proof

Ω

^′

= φ

⁻¹

]−∞, −δ[

, with δ > 0 small.

^(recall φ)

Observe:

by continuity, dφ 6= 0 in φ

⁻¹

[−δ, 0]

, so ∂ Ω

^′

is smooth;

Ω

^′

is strongly concave, by continuity of Hess(φ) and compactness of ∂ Ω ;

focal radius

if δ < foc(∂ Ω) , every OGC in Ω

^′

can be extended to an OGC in Ω

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Proof

Ω

^′

= φ

⁻¹

]−∞, −δ[

, with δ > 0 small.

^(recall φ)

Observe:

by continuity, dφ 6= 0 in φ

⁻¹

[−δ, 0]

, so ∂ Ω

^′

is smooth;

Ω

^′

is strongly concave, by continuity of Hess(φ) and compactness of ∂ Ω ;

if δ < foc(∂ Ω) , every OGC in Ω

^′

can be extended to an OGC in Ω

if by absurd ∃δ

_n

→ 0 and a sequence γ

_n

of WOGC’s in φ

⁻¹

]−∞, −δ

_n

[

, then one would get infinitely many

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The main geometrical result

Theorem. (Giambò, Giannoni, Piccione)

Let (M, g ) be a Riemannian manifold. Assume:

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The main geometrical result

Theorem. (Giambò, Giannoni, Piccione)

Let (M, g ) be a Riemannian manifold. Assume:

Ω ⊂ M open;

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The main geometrical result

Theorem. (Giambò, Giannoni, Piccione)

Let (M, g ) be a Riemannian manifold. Assume:

Ω ⊂ M open;

Ω homeomorphic to S

^m⁻¹

× [0, 1] ;

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The main geometrical result

Theorem. (Giambò, Giannoni, Piccione)

Let (M, g ) be a Riemannian manifold. Assume:

Ω ⊂ M open;

Ω homeomorphic to S

^m⁻¹

× [0, 1] ;

Ω strongly concave.

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The main geometrical result

Theorem. (Giambò, Giannoni, Piccione)

Let (M, g ) be a Riemannian manifold. Assume:

Ω ⊂ M open;

Ω homeomorphic to S

^m⁻¹

× [0, 1] ; Ω strongly concave.

Then, there are at least two (geometrically distinct) crossing OCG’s in Ω .

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The main geometrical result

Theorem. (Giambò, Giannoni, Piccione)

Let (M, g ) be a Riemannian manifold. Assume:

Ω ⊂ M open;

Ω homeomorphic to S

^m⁻¹

× [0, 1] ; Ω strongly concave.

Then, there are at least two (geometrically distinct) crossing OCG’s in Ω .

Obs.: Recall that it suffices to consider the case that there

are no WOGC’s.

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The main geometrical result

Theorem. (Giambò, Giannoni, Piccione)

Let (M, g ) be a Riemannian manifold. Assume:

Ω ⊂ M open;

Ω homeomorphic to S

^m⁻¹

× [0, 1] ; Ω strongly concave.

Then, there are at least two (geometrically distinct) crossing OCG’s in Ω .

Obs.: Recall that it suffices to consider the case that there are no WOGC’s.

Obs.: Again, the result is optimal. Recall

example above (with opposite strict inequalities!)).

back to central symmetry School in Nonlinear Analysis and Calculus of Variations – p. 11/68

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Central symmetry

Def.: (M, g ) Riemannian man., A ⊂ M is centrally

symmetric around x

₀

∈ M if exists an isometry I : M → M , with I

²

= I , whose unique fixed pt is x

₀

, and such that

I (A) = A.

A function f : M → R is centrally symmetric if f ◦ I = f .

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Central symmetry

Def.: (M, g ) Riemannian man., A ⊂ M is centrally

symmetric around x

₀

∈ M if exists an isometry I : M → M , with I

²

= I , whose unique fixed pt is x

₀

, and such that

I (A) = A.

A function f : M → R is centrally symmetric if f ◦ I = f .

If γ is a geodesic (orthogonal to Σ ), then I ◦ γ is a geodesic (orthogonal to I (Σ) )

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Central symmetry

Def.: (M, g ) Riemannian man., A ⊂ M is centrally

symmetric around x

₀

∈ M if exists an isometry I : M → M , with I

²

= I , whose unique fixed pt is x

₀

, and such that

I (A) = A.

A function f : M → R is centrally symmetric if f ◦ I = f .

If γ is a geodesic (orthogonal to Σ ), then I ◦ γ is a geodesic (orthogonal to I (Σ) )

Theorem. Under the assumptions of the above theorem, if

Ω is centrally symmetric around some x

₀

, then there are at

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A short history of the problem

Two classical results:

Ljusternik and Schnirelmann, 1932:

there are at least n principal chords in a compact

convex subset of the n -dimensional Euclidean space having C

²

boundary

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A short history of the problem

Two classical results:

Ljusternik and Schnirelmann, 1932:

there are at least n principal chords in a compact

convex subset of the n -dimensional Euclidean space having C

²

boundary

W. Bos, Kritische Sehenen auf Riemannischen Elementarraumstücken, Math. Ann. 1963

at least n OGC’s in convex Riemannian manifolds

homeomorphic to an n -disk.

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A short history of the problem

Two classical results:

Ljusternik and Schnirelmann, 1932:

there are at least n principal chords in a compact

convex subset of the n -dimensional Euclidean space having C

²

boundary

W. Bos, Kritische Sehenen auf Riemannischen Elementarraumstücken, Math. Ann. 1963

at least n OGC’s in convex Riemannian manifolds homeomorphic to an n -disk.

F. Giannoni, P. Majer, On the effect of the domain about the multiplicity of the orthogonal geodesic chords,

Diff. Geom. Appl. 1997

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The topology of the manifold

G & M’s result:

if the manifold is homeomorphic to an annulus and it is convex, then there are at least two OGC’s;

if the manifold has compact and convex boundary,

and if the LS-category of the space of paths with

endpoints on the boundary is infinite, then there are

infinitely many OGC’s.

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The topology of the manifold

G & M’s result:

if the manifold is homeomorphic to an annulus and it is convex, then there are at least two OGC’s;

if the manifold has compact and convex boundary, and if the LS-category of the space of paths with endpoints on the boundary is infinite, then there are infinitely many OGC’s.

←− Example

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Geodesics chords, homoclinics and brake orbits: a short bibliography

H. Seifert, Periodische Bewegungen Machanischer

Systeme, Math. Z. 1948.

(64)

Geodesics chords, homoclinics and brake orbits: a short bibliography

H. Seifert, Periodische Bewegungen Machanischer Systeme, Math. Z. 1948.

H. Gluck, W. Ziller, Existence of Periodic Motions of Conservative Systems, in “Seminar on Minimal

Surfaces” (E. Bombieri Ed.), 1983. (picture)

(65)

Geodesics chords, homoclinics and brake orbits: a short bibliography

H. Seifert, Periodische Bewegungen Machanischer Systeme, Math. Z. 1948.

H. Gluck, W. Ziller, Existence of Periodic Motions of Conservative Systems, in “Seminar on Minimal

Surfaces” (E. Bombieri Ed.), 1983. (picture)

A. Weinstein, Periodic orbits for convex Hamiltonian

systems, Ann. of Math. 1978.

(66)

Geodesics chords, homoclinics and brake orbits: a short bibliography

H. Seifert, Periodische Bewegungen Machanischer Systeme, Math. Z. 1948.

H. Gluck, W. Ziller, Existence of Periodic Motions of Conservative Systems, in “Seminar on Minimal

Surfaces” (E. Bombieri Ed.), 1983. (picture)

A. Weinstein, Periodic orbits for convex Hamiltonian systems, Ann. of Math. 1978.

These results will be reviewed later.

(67)

More bibliography

P. H. Rabinowitz, Periodic and Eteroclinic Orbits for a

Periodic Hamiltonian System, Ann. Inst. H. Poincaré

1989.

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More bibliography

P. H. Rabinowitz, Periodic and Eteroclinic Orbits for a Periodic Hamiltonian System, Ann. Inst. H. Poincaré 1989.

A. Ambrosetti, V. Coti Zelati, Multiple Homoclinic Orbits for a Class of Conservative Systems, Rend.

Sem. Mat. Univ. Padova, 1993.

(69)

More bibliography

P. H. Rabinowitz, Periodic and Eteroclinic Orbits for a Periodic Hamiltonian System, Ann. Inst. H. Poincaré 1989.

A. Ambrosetti, V. Coti Zelati, Multiple Homoclinic Orbits for a Class of Conservative Systems, Rend.

Sem. Mat. Univ. Padova, 1993.

K. Tanaka, A Note on the Existence of Multiple

Homoclinic Orbits for a Perturbed Radial Potential,

No. D. E. A. 1994.

(70)

More bibliography

P. H. Rabinowitz, Periodic and Eteroclinic Orbits for a Periodic Hamiltonian System, Ann. Inst. H. Poincaré 1989.

A. Ambrosetti, V. Coti Zelati, Multiple Homoclinic Orbits for a Class of Conservative Systems, Rend.

Sem. Mat. Univ. Padova, 1993.

K. Tanaka, A Note on the Existence of Multiple

Homoclinic Orbits for a Perturbed Radial Potential, No. D. E. A. 1994.

E. Paturel, Multiple homoclinic orbits for a class of Hamiltonian systems, Calc. Var. & PDE’s 2001.

(71)

More bibliography

P. H. Rabinowitz, Periodic and Eteroclinic Orbits for a Periodic Hamiltonian System, Ann. Inst. H. Poincaré 1989.

A. Ambrosetti, V. Coti Zelati, Multiple Homoclinic Orbits for a Class of Conservative Systems, Rend.

Sem. Mat. Univ. Padova, 1993.

K. Tanaka, A Note on the Existence of Multiple

Homoclinic Orbits for a Perturbed Radial Potential, No. D. E. A. 1994.

E. Paturel, Multiple homoclinic orbits for a class of

(72)

Bibliography for this course

R. Giambò, F. Giannoni, P. Piccione,

Orthogonal Geodesic Chords, Brake Orbits and Homoclinic Orbits in Riemannian Manifolds,

Advances in Diff. Eq. 2005.

On the multiplicity of brake orbits and homoclinics in Riemannian manifolds, Acc. Lincei, 2005.

Multiple brake orbits and homoclinics in Riemannian manifolds, preprint 2004.

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Ljusternik–Schnirelman category

Def.: X top. space, Y ⊂ X is contractible in X if i : Y → X is homotopic to a constant.

LS-category

cat_X(Y) = min

n : ∃C₁, . . . , C_n ⊂ X open and contractible, Y ⊂

n

[

k=1

C_k ∈ {0,1, . . . ,+∞}.

(74)

Ljusternik–Schnirelman category

LS-category

cat_X(Y) = min

n

[

k=1

C_k ∈ {0,1, . . . ,+∞}.

cat is homotopy invariant and monotonic increasing by inclusion.

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Ljusternik–Schnirelman category

LS-category

cat_X(Y) = min

n

[

k=1

C_k ∈ {0,1, . . . ,+∞}.

Classical result. If X is a complete Banach manifold and f : X → R is C¹, bounded from below, and satisfies (PS), then f has at least cat(X) critical points.

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Ljusternik–Schnirelman category

LS-category

cat_X(Y) = min

n

[

k=1

C_k ∈ {0,1, . . . ,+∞}.

More generally, cat gives a lower estimate on the number of fixed points of flows.

(fixed pts of the gradient flow of f=critical pts. of f)

(77)

Ljusternik–Schnirelman category

LS-category

cat_X(Y) = min

n

[

k=1

C_k ∈ {0,1, . . . ,+∞}.

In the case of Riemannian manifolds with convex boundary, one can use the shortening

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Ljusternik–Schnirelman category

LS-category

cat_X(Y) = min

n

[

k=1

C_k ∈ {0,1, . . . ,+∞}.

In the case of Riemannian manifolds with convex boundary, one can use the shortening flow on the space of curves lying inside the manifold, and whose endpoints are on the boundary.

In the concave case, the shortening flow is not well defined on such space.

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Abstract LS theory

We will reproduce the “ingredients” of the classical LS theory in a nonsmooth context:

e

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Abstract LS theory

a metric space M. consists of curves having image in an open neighborhood of Ω ∼= S^m−1 × [0,1], whose endpoints remain near ∂Ω.

e

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Abstract LS theory

a compact subset C⊂ M, homeomorphic to the sphere S^m−1.

e

(82)

Abstract LS theory

a compact subset C⊂ M, homeomorphic to the sphere S^m−1. a family H^e consisting of pairs (D, h), where D ⊂ Cis compact and

h : [0,1] × D → Mis a continuous map with h(0, x) = x for all x, and satisfying other properties that will be discussed ahead.

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Abstract LS theory

a functional F, that associates to each pair (D, h) ∈ H^e a real number F(D, h).

(84)

Abstract LS theory

We define a suitable notion of critical pt for F, in such a way that distinct critical values of F correspond to geometrically distinct OGC’s in Ω.

(85)

Abstract LS theory

We define a suitable notion of critical pt for F, in such a way that distinct critical values of F correspond to geometrically distinct OGC’s in Ω.

We prove two deformation lemmas for the sublevels of F, and we prove a (PS) condition

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Deformation Lemmas and critical pts

1DL: noncritical levels of F can be deformed by homotopies in H e into lower levels;

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Deformation Lemmas and critical pts

1DL: noncritical levels of F can be deformed by homotopies in H e into lower levels;

2DL: a similar deformation exists also for critical levels of

F , provided that suitable neighborhoods of the critical pts

are removed.

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Deformation Lemmas and critical pts

1DL: noncritical levels of F can be deformed by homotopies in H e into lower levels;

2DL: a similar deformation exists also for critical levels of F , provided that suitable neighborhoods of the critical pts are removed.

i = 1, 2 : Γ

_i

=

D ∈ C : cat(D) ≥ i , c

_i

= inf

D∈Γ_i (D,h)∈H^e

F (D, h) One then proves:

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Deformation Lemmas and critical pts

1DL: noncritical levels of F can be deformed by homotopies in H e into lower levels;

2DL: a similar deformation exists also for critical levels of F , provided that suitable neighborhoods of the critical pts are removed.

i = 1, 2 : Γ

_i

=

D ∈ C : cat(D) ≥ i , c

_i

= inf

F (D, h) One then proves:

c

_i

> 0 and c

_i

< +∞ ;

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Deformation Lemmas and critical pts

1DL: noncritical levels of F can be deformed by homotopies in H e into lower levels;

2DL: a similar deformation exists also for critical levels of F , provided that suitable neighborhoods of the critical pts are removed.

i = 1, 2 : Γ

_i

=

D ∈ C : cat(D) ≥ i , c

_i

= inf

F (D, h) One then proves:

c

_i

> 0 and c

_i

< +∞ ; c

₁

≤ c

₂

;

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Deformation Lemmas and critical pts

1DL: noncritical levels of F can be deformed by homotopies in H e into lower levels;

2DL: a similar deformation exists also for critical levels of F , provided that suitable neighborhoods of the critical pts are removed.

i = 1, 2 : Γ

_i

=

D ∈ C : cat(D) ≥ i , c

_i

= inf

F (D, h) One then proves:

c

_i

> 0 and c

_i

< +∞ ;

(92)

Deformation Lemmas and critical pts

1DL: noncritical levels of F can be deformed by homotopies in H e into lower levels;

2DL: a similar deformation exists also for critical levels of F , provided that suitable neighborhoods of the critical pts are removed.

i = 1, 2 : Γ

_i

=

D ∈ C : cat(D) ≥ i , c

_i

= inf

F (D, h) One then proves:

c

_i

> 0 and c

_i

< +∞ ; c

₁

≤ c

₂

;

each c

_i

is a critical value, by 1DL; c

₁

< c

₂

by 2DL.

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Basic notations

Ω ∼= S^m−1 × [0,1] strongly concave, Ω ⊂ R^m, D₁, D₂ ∼= S^m−1 conn. comp. of ∂Ω.

Z

(s) γ(a)

γ(b) γ

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Basic notations

x ∈ H¹ [a, b],R^m

, kxka,b =

1

2 kx(a)k +

Z b

a kx(s)˙ k² ds

1 2

, kxkL^∞ ≤ kxka,b .

(s) γ(a)

γ(b) γ

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Basic notations

x ∈ H¹ [a, b],R^m

, kxka,b =

1

2 kx(a)k +

Z b

a kx(s)˙ k² ds

1 2

, kxkL^∞ ≤ kxka,b . φ : R^m → R, Ω = φ⁻¹ ]−∞,0[

, ∂Ω = φ⁻(0), dφ 6= 0 on ∂Ω and Hess(φ)[v, v] < 0 for v ∈ T(∂Ω) \ {0}.

(s) γ(a)

γ(b) γ

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Basic notations

x ∈ H¹ [a, b],R^m

, kxka,b =

1

2 kx(a)k +

Z b

a kx(s)˙ k² ds

1 2

∃δ₀ > 0 such that Hess(φ)_x[v, v] < 0 for all x ∈ φ⁻¹ [−δ₀, δ₀]

and all v 6= 0 with dφ_x[v] = 0.

(s) γ(a)

γ(b) γ

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Basic notations

x ∈ H¹ [a, b],R^m

, kxka,b =

1

2 kx(a)k +

Z b

a kx(s)˙ k² ds

1 2

ρ₀ = min

x∈D1

y∈D2

dist(x, y), K₀ = max

φ⁻¹ ]−∞,δ0]

k∇φk < +∞.

(s) γ(a)

γ(b) γ

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Basic notations

x ∈ H¹ [a, b],R^m

, kxka,b =

1

2 kx(a)k +

Z b

a kx(s)˙ k² ds

1 2

ρ₀ = min

x∈D1

y∈D2

dist(x, y), K₀ = max

φ⁻¹ ]−∞,δ0]

k∇φk < +∞.

(s) γ(a)

γ(b)

γ Prop.: If γ : [a, b] → Ω is a geo with γ(a), γ(b) ∈ ∂Ω, then ∃s¯ ∈ ]a, b[

with φ(γ(¯s)) < −δ₀.

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The path spaces

Define C_i = connected components of φ⁻¹ [0, δ₀]

containing D_i, i = 1,2.

M=

n

x ∈ H¹ [0,1],R^m

: φ(x(s)) < δ₀, x(0) ∈ C₁, x(1) ∈ C₂

o R

n

o

n o

R

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The path spaces

M=

n

x ∈ H¹ [0,1],R^m

: φ(x(s)) < δ₀, x(0) ∈ C₁, x(1) ∈ C₂

o R

n

o

n o

R

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The path spaces

M=

n

x ∈ H¹ [0,1],R^m

: φ(x(s)) < δ₀, x(0) ∈ C₁, x(1) ∈ C₂

o

C∼=

n

orthogonal segments from D₁ to D₂

o

M₀ = sup

x∈C

R 1

0 g( ˙x,x) ds˙ .

n o

R

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The path spaces

M=

n

x ∈ H¹ [0,1],R^m

: φ(x(s)) < δ₀, x(0) ∈ C₁, x(1) ∈ C₂

o

C∼=

n

o

M₀ = sup

x∈C

R 1

0 g( ˙x,x) ds˙ . x ∈ M, Jx⁰ =

n

[a, b] ⊂ [0,1] : x [a, b]

⊂ Ω, x(a) ∈ D₁, x(b) ∈ D₂

o

crossing intervals J^x =

n

[a, b] ∈ Jx⁰ : [a, b] maximal

o

R

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The path spaces

M=

n

x ∈ H¹ [0,1],R^m

: φ(x(s)) < δ₀, x(0) ∈ C₁, x(1) ∈ C₂

o

C∼=

n

o

M₀ = sup

x∈C

R 1

0 g( ˙x,x) ds˙ . x ∈ M, Jx⁰ =

n

[a, b] ⊂ [0,1] : x [a, b]

⊂ Ω, x(a) ∈ D₁, x(b) ∈ D₂

o

crossing intervals

n

0

o R

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The path spaces

M=

n

x ∈ H¹ [0,1],R^m

: φ(x(s)) < δ₀, x(0) ∈ C₁, x(1) ∈ C₂

o

C∼=

n

o

M₀ = sup

x∈C

R 1

0 g( ˙x,x) ds˙ . x ∈ M, Jx⁰ =

n

[a, b] ⊂ [0,1] : x [a, b]

⊂ Ω, x(a) ∈ D₁, x(b) ∈ D₂

o

crossing intervals J^x =

n

[a, b] ∈ Jx⁰ : [a, b] maximal

o

Def.: [a, b] ∈ Jx⁰ is an M₀-interval if ¹₂

Rb

a g( ˙x,x) ds < M˙ ₀. Obs.: x ∈ M=⇒ |J^x| < +∞: [a, b] ∈ Jx⁰, b − a ≥ ρ²₀

Rb

a g( ˙x,x) ds˙

−1

.

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Geometrically and variationally critical points

Def.: c ∈ ]0, M₀[ is a geometrically critical value if ∃ a crossing OGC γ : [0,1] → Ω with

1 2

R 1

0 g( ˙γ,γ) dt˙ = c. A geometrically regular value is a number c which is not geometrically critical.

n

o

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Geometrically and variationally critical points

1 2

R 1

Prop.: If c₁ 6= c₂ are GCV’s, then they correspond to geometrically distinct OGC’s.

n

o

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Geometrically and variationally critical points

1 2

R 1

V⁺(x) =

n

V vector field along x : g V (s),∇φ(x(s))

≥ 0 when x(s) ∈ φ⁻¹ [0, ^δ₂⁰]

o

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Geometrically and variationally critical points

1 2

R 1

V⁺(x) =

n

≥ 0 when x(s) ∈ φ⁻¹ [0, ^δ₂⁰]

o

V⁻(x) =

n

≤ 0 when x(s) ∈ φ⁻¹(0)

o