Functions on the sphere with critical points in pairs and orthogonal geodesic chords

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Functions on the sphere with critical points in pairs and orthogonal geodesic chords

RISM4 – Nonlinear Phenomena in Mathematics and Economics

Paolo Piccione

Universidade São Paulo, Brazil

September 16th, 2015

P. Piccione — USP, Brazil Functions on the sphere with critical points in pairs

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My co-authors

This is a joint work with:

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Outline of this talk.

Main topics

1 Topology: Morse-even functionson the sphere

2 ODE’s: brake orbitsfor conservative Lagrangian systems

– only as motivation for part (1) and (3)

3 Geometry: orthogonal geodesic chords

Abstract

I will discuss a problem of multiplicity for geodesics starting and arriving

orthogonally to the boundary of a Riemannian ball using Morse theory. This gives an analogous multiplicity result for a class of periodic solutions (brake orbits) in a potential well of a Lagrangian system.

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Outline of this talk.

Main topics

Abstract

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Outline of this talk.

Main topics

Abstract

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Outline of this talk.

Main topics

Abstract

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Morse even functions

The setup:

M^m is a compact manifold;

βk(M)denotes thek-th Betti number ofM,k =0, . . . ,m;

f:M→Ris a Morse function;

ifp∈Mis a critical point off,i_Morse(f,p)is the Morse index;

µ_k(f)is the number of critical pts off having Morse index equal tok

Definition

f isMorse-evenifµk(f)is even for allk =0, . . . ,m.

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Morse even functions

The setup:

M^m is a compact manifold;

βk(M)denotes thek-th Betti number ofM,k =0, . . . ,m;

f:M→Ris a Morse function;

ifp∈Mis a critical point off,i_Morse(f,p)is the Morse index;

µ_k(f)is the number of critical pts off having Morse index equal tok

Definition

f isMorse-evenifµ (f)is even for allk =0, . . . ,m.

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Morse-even functions on the sphere

Proposition

Iff:S^m→Ris Morse-even, thenµk(f)>0 for allk =0, . . . ,m.

Proof.

β0(S^m) =βm(S^m) =1,βk(S^m) =0.Morse relations: µ₀ ≥ β₀

µ₁−µ₀ ≥ β₁−β₀ . . .

µm−µm−1+. . .+ (−1)^mµ₀ ≥ βm−βm−1+. . .+ (−1)^mβ₀

µ0≥2.

µ₁≥µ₀+β₁−β₀=µ₀−1≥1.

µ₂≥µ₁−µ₀+β₂−β₁+β₀=µ₁−µ₀+1>0....

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Morse-even functions on the sphere

Proposition

Proof.

β0(S^m) =βm(S^m) =1,βk(S^m) =0.

Morse relations:

µ₀ ≥ β₀ µ₁−µ₀ ≥ β₁−β₀

. . .

µm−µm−1+. . .+ (−1)^mµ₀ ≥ βm−βm−1+. . .+ (−1)^mβ₀

µ0≥2.

µ₁≥µ₀+β₁−β₀=µ₀−1≥1.

µ₂≥µ₁−µ₀+β₂−β₁+β₀=µ₁−µ₀+1>0....

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Morse-even functions on the sphere

Proposition

Proof.

β0(S^m) =βm(S^m) =1,βk(S^m) =0. Morse relations:

µ₀ ≥ β₀ µ₁−µ₀ ≥ β₁−β₀

. . .

µm−µm−1+. . .+ (−1)^mµ₀ ≥ βm−βm−1+. . .+ (−1)^mβ₀

µ0≥2.

µ₁≥µ₀+β₁−β₀=µ₀−1≥1.

µ₂≥µ₁−µ₀+β₂−β₁+β₀=µ₁−µ₀+1>0....

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Morse-even functions on the sphere

Proposition

Proof.

µ₀ ≥ β₀ µ₁−µ₀ ≥ β₁−β₀

. . .

µm−µm−1+. . .+ (−1)^mµ₀ ≥ βm−βm−1+. . .+ (−1)^mβ₀

µ ≥2.

µ₁≥µ₀+β₁−β₀=µ₀−1≥1.

µ₂≥µ₁−µ₀+β₂−β₁+β₀=µ₁−µ₀+1>0....

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Morse-even functions on the sphere

Proposition

Proof.

µ₀ ≥ β₀ µ₁−µ₀ ≥ β₁−β₀

. . .

µm−µm−1+. . .+ (−1)^mµ₀ ≥ βm−βm−1+. . .+ (−1)^mβ₀

µ0≥2.

µ₁≥µ₀+β₁−β₀=µ₀−1≥1.

µ₂≥µ₁−µ₀+β₂−β₁+β₀=µ₁−µ₀+1>0....

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Morse-even functions on the sphere

Proposition

Proof.

µ₀ ≥ β₀ µ₁−µ₀ ≥ β₁−β₀

. . .

µm−µm−1+. . .+ (−1)^mµ₀ ≥ βm−βm−1+. . .+ (−1)^mβ₀

µ ≥2.

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Generalization to Morse even functions on arbitrary manifolds

Theorem

If M^m is a compact manifold which is connected and orientable (β₀(M) =βm(M) =1) withβk(M)∈2Nfor all k =1, . . . ,m−1, and f:M→Ris a Morse-even function, then:

µk(f)> βk, for all k =0, . . . ,m.

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Classification of low dimensional manifolds with even Betti numbers

Problem: Classify compact connected orientablen-manifolds Mwithβ_k(M)even for allk =1, . . . ,n−1.

n=2 for every compact orientable surfaceM²of genusg: β₁(M) =2g

n=3 the only compact simply connectedM³isS³ n=4 Recall thatβ2isadditive by connected sum: β₂(M₁⁴]M₂⁴) =β₂(M₁⁴) +β₂(M₂⁴)

β₂(S²×S²) =2, β₂(CP²) =1

M = (S²×S²)]· · ·](S²×S²)

| {z }

ktimes

] CP²]· · ·]CP²

| {z }

(2m)times

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n=2 for every compact orientable surfaceM²of genusg:

β₁(M) =2g

n=3 the only compact simply connectedM³isS³ n=4 Recall thatβ2isadditive by connected sum: β₂(M₁⁴]M₂⁴) =β₂(M₁⁴) +β₂(M₂⁴)

β₂(S²×S²) =2, β₂(CP²) =1

M = (S²×S²)]· · ·](S²×S²)

| {z }

ktimes

] CP²]· · ·]CP²

| {z }

(2m)times

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β₁(M) =2g

n=3 the only compact simply connectedM³isS³

n=4 Recall thatβ2isadditive by connected sum: β₂(M₁⁴]M₂⁴) =β₂(M₁⁴) +β₂(M₂⁴)

β₂(S²×S²) =2, β₂(CP²) =1

M = (S²×S²)]· · ·](S²×S²)

| {z }

ktimes

] CP²]· · ·]CP²

| {z }

(2m)times

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β₁(M) =2g

n=3 the only compact simply connectedM³isS³ n=4 Recall thatβ2isadditive by connected sum:

β₂(M₁⁴]M₂⁴) =β₂(M₁⁴) +β₂(M₂⁴)

β₂(S²×S²) =2, β₂(CP²) =1

M = (S²×S²)]· · ·](S²×S²)

| {z }

ktimes

] CP²]· · ·]CP²

| {z }

(2m)times

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β₁(M) =2g

β₂(M₁⁴]M₂⁴) =β₂(M₁⁴) +β₂(M₂⁴)

β₂(S²×S²) =2, β₂(CP²) =1

M = (S²×S²)]· · ·](S²×S²)

| {z }

ktimes

] CP²]· · ·]CP²

| {z }

(2m)times

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β₁(M) =2g

β₂(M₁⁴]M₂⁴) =β₂(M₁⁴) +β₂(M₂⁴)

β₂(S²×S²) =2, β₂(CP²) =1

M = (S²×S²)]· · ·](S²×S²)

| {z }

ktimes

] CP²]· · ·]CP²

| {z }

(2m)times

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ODE’s: periodic solutions of Lagrangian systems

Conservative Lagrangian systems:

(M,g)Riemannian manifold (configuration space) V:M →Rpotential function (dynamics)

Lagrangian system: _dt^Dx˙ =−∇V(x)

Energyof a solution:E = ¹₂g( ˙x,x˙) +V(x)

We look forperiodicsolutions with given energyE. Maupertuis’ Principle

Solutions of energyE ⇐⇒

geodesics inΩ_E =V⁻¹ ]−∞,E] relatively to the conformal metric gE = (E−V)·g

Obs.:g_E issingularon∂Ω_E =V⁻¹(E).

Special class of periodic solutions:brake orbits(pendulum-like)

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ODE’s: periodic solutions of Lagrangian systems

Lagrangian system: _dt^Dx˙ =−∇V(x) Energyof a solution:E = ¹₂g( ˙x,x˙) +V(x)

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ODE’s: periodic solutions of Lagrangian systems

We look forperiodicsolutions with given energyE.

Maupertuis’ Principle Solutions of energyE ⇐⇒

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ODE’s: periodic solutions of Lagrangian systems

geodesics inΩ_E =V⁻¹ ]−∞,E]

relatively to the conformal metric gE = (E−V)·g

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ODE’s: periodic solutions of Lagrangian systems

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ODE’s: periodic solutions of Lagrangian systems

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Seifert’s conjecture (1947)

Conjecture Assume:

E regular value ofV;

Ω_E is homeomorphic to the ballB^m+1.

Then, there are at leastm+1distinct brake orbits of energyE. Problem: needm+1 geodesics with endpoints on the singular boundary

Geometric construction:

remove fromΩ_E a suitably defined neighborhoodV of∂Ω_E; geodesics inΩ_E with endpoints in∂Ω_E correspond to geodesics inΩ⁰= Ω_E \V arriving orthogonally to∂Ω⁰ Ω⁰ is homeomorphic toΩ_E ∼=B^m+1

∂Ω⁰∼=S^misconcave.

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Seifert’s conjecture (1947)

Conjecture Assume:

Then, there are at leastm+1distinct brake orbits of energyE. Problem: needm+1 geodesics with endpoints on the singular boundary

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Seifert’s conjecture (1947)

Conjecture Assume:

Then, there are at leastm+1distinct brake orbits of energyE.

Problem: needm+1 geodesics with endpoints on the singular boundary

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Seifert’s conjecture (1947)

Conjecture Assume:

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Seifert’s conjecture (1947)

Conjecture Assume:

remove fromΩ_E a suitably defined neighborhoodV of∂Ω_E; geodesics inΩ_E with endpoints in∂Ω_E correspond to

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Geometry: orthogonal geodesic chords

(Ω,g)compact Riemannian manifold with boundary∂Ω γ: [0,T]−→Ω geodesic withγ(0), γ(T)∈∂Ω

γ is anorthogonal geodesic chordifγ(0),˙ γ(T˙ )∈T(∂Ω)^⊥ νunit normal vector field along∂Ωpointing insideΩ.

p∈∂Ω,γp(t) =exp_p(t·νp)

Basic assumptions on(Ω,g)

For allp∈∂Ω:

(HP1) ∃T_p>0 such that:

γ_p(t)6∈∂Ωfort ∈]0,T_p[;

γ_pmeets∂Ωtransversallyatt =T_p. (HP2) γp(Tp)isnotafocal pointalongγp.

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Geometry: orthogonal geodesic chords

(Ω,g)compact Riemannian manifold with boundary∂Ω

γ: [0,T]−→Ω geodesic withγ(0), γ(T)∈∂Ω

For allp∈∂Ω:

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Geometry: orthogonal geodesic chords

For allp∈∂Ω:

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Geometry: orthogonal geodesic chords

γ is anorthogonal geodesic chordifγ(0),˙ γ(T˙ )∈T(∂Ω)^⊥

νunit normal vector field along∂Ωpointing insideΩ. p∈∂Ω,γp(t) =exp_p(t·νp)

For allp∈∂Ω:

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Geometry: orthogonal geodesic chords

For allp∈∂Ω:

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Geometry: orthogonal geodesic chords

For allp∈∂Ω:

γ meets∂Ωtransversallyatt =T .

(HP2) γp(Tp)isnotafocal pointalongγp.

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Geometry: orthogonal geodesic chords

For allp∈∂Ω:

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How bad are the assumptions?

(HP1) is anopen conditionrelatively to theC¹-topology (HP2) is anopen conditionrelatively to theC²-topology Radially symmetricmetrics on balls satisfy (HP1) and (HP2)

Neither (HP1) nor (HP2) isgeneric.

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How bad are the assumptions?

(HP1) is anopen conditionrelatively to theC¹-topology

(HP2) is anopen conditionrelatively to theC²-topology Radially symmetricmetrics on balls satisfy (HP1) and (HP2)

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How bad are the assumptions?

(HP1) is anopen conditionrelatively to theC¹-topology (HP2) is anopen conditionrelatively to theC²-topology

Radially symmetricmetrics on balls satisfy (HP1) and (HP2)

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How bad are the assumptions?

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How bad are the assumptions?

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Critical points of the crossing time

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Obs. 1:By transversality (HP1), T:∂Ω−→]0,+∞[is smooth.

Crossing timefunction of(Ω,g). Theorem

Under assumption (HP2), p is a critical point of T iffγ_pis an orthogonal geodesic chord, i.e., iff

˙

γp(Tp)⊥∂Ω.

Obs. 2:Critical points of T :∂Ω→Rcome in pairs!

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Critical points of the crossing time

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Crossing timefunction of(Ω,g).

Theorem

˙

γp(Tp)⊥∂Ω.

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Critical points of the crossing time

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Theorem

˙

γp(Tp)⊥∂Ω.

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Critical points of the crossing time

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Theorem

˙

γp(Tp)⊥∂Ω.

Obs. 2:Critical points of

→

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Morse indices associated to an OGC

γ_p: [0,T_p]−→Ωorthogonal geodesic chord.

γp(0) =p,γp(T_p) =q

γq=γ_p⁻(backward reparameterization)

3 different notions of Morse index associated toγ

1 Morse index ofγpas a free endpoints geodesic: i_free(γp)

2 Morse index ofγ_pas fixed endpoint geodesic: i_fixed(γ_p)

3 Morse index of the crossing time: i_Morse(T,p) Theorem

(a) i_fixed(γ_p)equals the number of∂Ω-focal pts along γ_p. (b) i_free(γp) =i_fixed(γp) +i_Morse(T,p)

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Morse indices associated to an OGC

γq=γ_p⁻(backward reparameterization) 3 different notions of Morse index associated toγ

3 Morse index of the crossing time: i_Morse(T,p) Theorem

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Morse indices associated to an OGC

3 Morse index of the crossing time:i_Morse(T,p)

Theorem

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Morse indices associated to an OGC

3 Morse index of the crossing time:i_Morse(T,p) Theorem

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The crossing time is a Morse-even function

In general, the number focal points depends on the orientation!

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Theorem

Under assumption (HP2), T is a

Morse-even function.

Proof.

Stability of the focal points They do not collapse onto the boundary

Obs.:Example shows that (HP2) isnot generic.

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The crossing time is a Morse-even function

W

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Theorem

Proof.

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The crossing time is a Morse-even function

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Theorem

Proof.

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The crossing time is a Morse-even function

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Theorem

Proof.

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Main Result

Theorem

Let g be a metric on B^m+1satisfying (HP1) and (HP2). Then, there are at least m+1distinct orthogonal geodesic chords in B^m.

This settles Seifert’s conjecture in aquite largenumber of cases.

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Main Result

Theorem

Let g be a metric on B^m+1satisfying (HP1) and (HP2).

Then, there are at least m+1distinct orthogonal geodesic chords in B^m.

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Main Result

Theorem

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Main Result

Theorem

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Counterexample when ∂ Ω is not a sphere

When∂Ωis not connected, one cannot expect the existence of more than2 OGC’s, regardless of the dimension.

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Γ₁

Γ₂