Bifurcation and symmetry breaking in geometric variational problems

Bifurcation and symmetry breaking in geometric variational problems

Joint work with M. Koiso and B. Palmer

Paolo Piccione

Departamento de Matemática Instituto de Matemática e Estatística

Universidade de São Paulo

EIMAN

I ENCONTRO INTERNACIONAL DE MATEMÁTICA

Generalities on variational bifurcation

General bifurcation setup:

Mdifferentiable manifold (possiblydim=∞)

f_λ:M→Rfamily of smooth functionals,λ∈[a,b]

λ7→x_λ ∈Msmooth curve of critical points: dfλ(x_λ) =0 for allλ.

Definition Bifurcationat

λ₀∈]a,b[if∃λ_n →λ₀ andx_n →x_λ0 as n→ ∞, with:

(a) dfλn(x_n) =0 for all n;

(b) x_n6=x_λn for alln.

Generalities on variational bifurcation

General bifurcation setup:

Mdifferentiable manifold (possiblydim=∞) f_λ:M→Rfamily of smooth functionals,λ∈[a,b]

λ7→x_λ ∈Msmooth curve of critical points: dfλ(x_λ) =0 for allλ.

Definition Bifurcationat

λ₀∈]a,b[if∃λ_n →λ₀ andx_n →x_λ0 as n→ ∞, with:

(a) dfλn(x_n) =0 for all n;

(b) x_n6=x_λn for alln.

Generalities on variational bifurcation

General bifurcation setup:

Mdifferentiable manifold (possiblydim=∞) f_λ:M→Rfamily of smooth functionals,λ∈[a,b]

λ7→x_λ ∈Msmooth curve of critical points: dfλ(x_λ) =0 for allλ.

Definition Bifurcationat

λ₀∈]a,b[if∃λ_n →λ₀ andx_n →x_λ0 as n→ ∞, with:

(a) dfλn(x_n) =0 for all n;

(b) x_n6=x_λn for alln.

Generalities on variational bifurcation

General bifurcation setup:

Mdifferentiable manifold (possiblydim=∞) f_λ:M→Rfamily of smooth functionals,λ∈[a,b]

λ7→x_λ ∈Msmooth curve of critical points: dfλ(x_λ) =0 for allλ.

Definition Bifurcationat

λ₀∈]a,b[if∃λ_n →λ₀ andx_n →x_λ0 as n→ ∞, with:

(a) dfλn(x_n) =0 for all n;

Generalities on variational bifurcation

General bifurcation setup:

Mdifferentiable manifold (possiblydim=∞) f_λ:M→Rfamily of smooth functionals,λ∈[a,b]

λ7→x_λ ∈Msmooth curve of critical points: dfλ(x_λ) =0 for allλ.

Definition Bifurcationat

λ₀∈]a,b[if∃λ_n →λ₀ andx_n →x_λ0 as n→ ∞, with:

(a) dfλn(x_n) =0 for all n;

(b) x_n6=x_λn for alln.

Equivariant bifurcation

Assume:

GLie group acting onM f_λisG-invariant for allλ

Note: the orbitG·x_λ consists of critical points.

Definition

Orbit bifurcationatλ₀∈]a,b[if∃λ_n→λ₀andx_n →x_λ0 as n→ ∞, with:

(a) df_λn(x_n) =0 for alln; (b) G·x_n 6=G·x_λn for alln.

Standard bifurcation theory requires a quite involved variational setup: differentiability, Palais–Smale, Fredholmness...

Bifurcation occurs atdegeneratecritical points withjumpsof the Morse index. In the equivariant case, bifurcation occurs at degenerate critical orbits wherejumps of thecritical groups.

Equivariant bifurcation

Assume:

GLie group acting onM f_λisG-invariant for allλ

Note: the orbitG·x_λ consists of critical points.

Definition

Orbit bifurcationatλ₀∈]a,b[if∃λ_n→λ₀andx_n→x_λ0 as n→ ∞, with:

(a) df_λn(x_n) =0 for alln;

(b) G·x_n6=G·x_λn for alln.

Standard bifurcation theory requires a quite involved variational setup: differentiability, Palais–Smale, Fredholmness...

Bifurcation occurs atdegeneratecritical points withjumpsof the Morse index. In the equivariant case, bifurcation occurs at degenerate critical orbits wherejumps of thecritical groups.

Equivariant bifurcation

Assume:

GLie group acting onM f_λisG-invariant for allλ

Note: the orbitG·x_λ consists of critical points.

Definition

Orbit bifurcationatλ₀∈]a,b[if∃λ_n→λ₀andx_n→x_λ0 as n→ ∞, with:

(a) df_λn(x_n) =0 for alln;

(b) G·x_n6=G·x_λn for alln.

Standard bifurcation theory requires a quite involved variational setup: differentiability, Palais–Smale, Fredholmness...

Bifurcation occurs atdegeneratecritical points withjumpsof the Morse index. In the equivariant case, bifurcation occurs at degenerate critical orbits wherejumps of thecritical groups.

Equivariant bifurcation

Assume:

GLie group acting onM f_λisG-invariant for allλ

Note: the orbitG·x_λ consists of critical points.

Definition

Orbit bifurcationatλ₀∈]a,b[if∃λ_n→λ₀andx_n→x_λ0 as n→ ∞, with:

(a) df_λn(x_n) =0 for alln;

(b) G·x_n6=G·x_λn for alln.

Standard bifurcation theory requires a quite involved variational setup: differentiability, Palais–Smale, Fredholmness...

Bifurcation occurs atdegeneratecritical points withjumpsof the Morse index. In the equivariant case, bifurcation occurs at degenerate critical orbits wherejumps of thecritical groups.

Constant mean curvature variational problem

M^m compact oriented manifold

(Nⁿ,g)oriented Riemannian manifold n=m+1

x : M , → N embedding

Mean curvature:Hx =tr 2^nd fund. form

Variational principle

x hasconstantmean curvature (CMC) iffx is a stationary point for thearea functionalrestricted to embeddings of fixedvolume.

Constant mean curvature variational problem

M^m compact oriented manifold (Nⁿ,g)oriented Riemannian manifold

n=m+1

x : M , → N embedding

Mean curvature:Hx =tr 2^nd fund. form

Variational principle

x hasconstantmean curvature (CMC) iffx is a stationary point for thearea functionalrestricted to embeddings of fixedvolume.

Constant mean curvature variational problem

M^m compact oriented manifold (Nⁿ,g)oriented Riemannian manifold n=m+1

x : M , → N embedding

Mean curvature:Hx =tr 2^nd fund. form

Variational principle

x hasconstantmean curvature (CMC) iffx is a stationary point for thearea functionalrestricted to embeddings of fixedvolume.

Constant mean curvature variational problem

M^m compact oriented manifold (Nⁿ,g)oriented Riemannian manifold n=m+1

x : M , → N embedding

Mean curvature:Hx =tr 2^nd fund. form

Variational principle

x hasconstantmean curvature (CMC) iffx is a stationary point for thearea functionalrestricted to embeddings of fixedvolume.

Constant mean curvature variational problem

M^m compact oriented manifold (Nⁿ,g)oriented Riemannian manifold n=m+1

x : M , → N embedding

Mean curvature:Hx =tr 2^nd fund. form

Variational principle

x hasconstantmean curvature (CMC) iffx is a stationary point for thearea functionalrestricted to embeddings of fixedvolume.

Constant mean curvature variational problem

M^m compact oriented manifold (Nⁿ,g)oriented Riemannian manifold n=m+1

x : M , → N embedding

Mean curvature:Hx =tr 2^nd fund. form

Variational principle

x hasconstantmean curvature (CMC) iffx is a stationary point for thearea functionalrestricted to embeddings of fixedvolume.

The nodary

A portion of nodoid, with boundary on parallel planes orthogonal to

the axis.

The nodary

The nodary and the symmetry axis.

A portion of nodoid, with boundary on parallel planes orthogonal to

the axis.

Fixed boundary problem

Circles for the CMC fixed boundary problem, lying on the planesΠ₀andΠ₁. In the middle, the symmetry planeΠ.

The 1-parameter family of nodoids

Σa,H,t₀ surface of revolution around thex₃-axis with generatrix thenodary:

x₁(t) = cost+√

cos²t+a

2|H| , t∈[−t₀,t₀]

x₃(t) = 1 2|H|

Z t

0

cosτ +√

cos²τ +a

√cos²τ+a cosτdτ

H= mean curvature

a=2cHfrom conservation law: 2x₁cost+2Hx₁²=c

proposition

There existreal-analyticfunctionsa=a(t₀)andH=H(t₀) such thatΣt₀ = Σ_{a(t0),H(t0),t0} satisfies the boundary condition.

The 1-parameter family of nodoids

Σa,H,t₀ surface of revolution around thex₃-axis with generatrix thenodary:

x₁(t) = cost+√

cos²t+a

2|H| , t∈[−t₀,t₀]

x₃(t) = 1 2|H|

Z t

0

cosτ +√

cos²τ +a

√cos²τ+a cosτdτ

H= mean curvature

a=2cHfrom conservation law: 2x₁cost+2Hx₁²=c

proposition

There existreal-analyticfunctionsa=a(t₀)andH=H(t₀) such thatΣt₀ = Σ_{a(t0),H(t0),t0} satisfies the boundary condition.

The 1-parameter family of nodoids

Σa,H,t₀ surface of revolution around thex₃-axis with generatrix thenodary:

x₁(t) = cost+√

cos²t+a

2|H| , t∈[−t₀,t₀]

x₃(t) = 1 2|H|

Z t

0

cosτ +√

cos²τ +a

√cos²τ+a cosτdτ

H= mean curvature

a=2cHfrom conservation law: 2x₁cost+2Hx₁²=c

proposition

There existreal-analyticfunctionsa=a(t₀)andH =H(t₀) such thatΣt₀ = Σ_{a(t0),H(t0),t0} satisfies the boundary condition.

a = a(t

)

3 6 9 12

a

Nodaries through two circles

-r0 r0

x

-h₀ 2 h₀ 2

x

Nodary curves that generate nodoids which pass through 2 circles. The bifurcation point is in the middle (thicker/red), it has horizontal tangent at the point of intersection with the circles.

The inner circle is a limit of the family whena→0.

The Jacobi operator

Jf =−∆f −(k₁²+k₂²)f Eigenvaluesλ1< λ2< ...→+∞

Courant’s nodal domain theorem

Jf =λ_kf =⇒ f has at leastk nodal domains Separation of variables:f =T(θ)·S(s)

T⁰⁰+κT =0, T(0) =T(2π), T⁰(0) =T⁰(2π),

−(xS⁰)⁰+ κ

x −x(k₁²+k₂²)

S =λxS, S −^L₂

=S ^L₂

=0.

Spherical caps with the same boundary

Degenerate nodoids

Degenerate nodoids are tangent to the planes containing their boundary. On the left, nodoids from the familyΣ, on the right

Degenerate nodoid with one bulge

Two nodal domains

Six nodal domains

The degenerate nodoid Σ

Bifurcating branch of nodoids at the instant t

= π

Position vector

x ₃

A picture of Miyuki and Bennett

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