A scheme for the proof of main result

A scheme for the proof of main result

I Introduce a setEmb(M,N)of embeddingsM ,→N.

(which regularity?)

I Manifold structure on the quotientM=Emb(M,N)/Diff(M).

I Consider the action of the isometry groupG=Iso(N,g)onM.

I Volume and area functionals onM(invariant byG).

CMC embeddings

M ,→N

!

constrained critical points of Area with fixed Volume.

Accumulation of non congruent

CMC embeddings

!

Constrained critical

G-orbit bifurcation

A scheme for the proof of main result

I Introduce a setEmb(M,N)of embeddingsM ,→N.

(which regularity?)

I Manifold structure on the quotientM=Emb(M,N)/Diff(M).

I Consider the action of the isometry groupG=Iso(N,g)onM.

I Volume and area functionals onM(invariant byG).

CMC embeddings

M ,→N

!

constrained critical points of Area with fixed Volume.

Accumulation of non congruent

CMC embeddings

!

Constrained critical

G-orbit bifurcation

A scheme for the proof of main result

I Introduce a setEmb(M,N)of embeddingsM ,→N.

(which regularity?)

I Manifold structure on the quotientM=Emb(M,N)/Diff(M).

I Consider the action of the isometry groupG=Iso(N,g)onM.

I Volume and area functionals onM(invariant byG).

CMC embeddings

M ,→N

!

constrained critical points of Area with fixed Volume.

Accumulation of non congruent

CMC embeddings

!

Constrained critical

G-orbit bifurcation

A scheme for the proof of main result

I Introduce a setEmb(M,N)of embeddingsM ,→N.

(which regularity?)

I Manifold structure on the quotientM=Emb(M,N)/Diff(M).

I Consider the action of the isometry groupG=Iso(N,g)onM.

I Volume and area functionals onM(invariant byG).

CMC embeddings

M ,→N

!

constrained critical points of Area with fixed Volume.

Accumulation of non congruent

CMC embeddings

!

Constrained critical

G-orbit bifurcation

A set of axioms for equivariant constrained bifurcation — 1

(A1) Msmooth manifold modeled on a separable Banach spaceX;

(A2) Gcompact (connected) Lie group actiongcontinuouslyonM; (A3) A:M→RsmoothG-invariant function;

(A4) V :M→RsmoothG-invariant function without critical points; (A5) orbits of critical points off_λ=A+λ· Vsmoooth submanifolds of

M.

In the CMC variational problem:

I M=Emb(M,N)/Diff(M)

I X= space of sections of some vector bundle overM

I G(connected component of 1 of) isometry group of(N,g)

I A= area functional, V= volume functional

I λ= mean curvature (up to a factor)

A set of axioms for equivariant constrained bifurcation — 1

(A1) Msmooth manifold modeled on a separable Banach spaceX;

(A2) Gcompact (connected) Lie group actiongcontinuouslyonM;

(A3) A:M→RsmoothG-invariant function;

(A4) V :M→RsmoothG-invariant function without critical points; (A5) orbits of critical points off_λ=A+λ· Vsmoooth submanifolds of

M.

In the CMC variational problem:

I M=Emb(M,N)/Diff(M)

I X= space of sections of some vector bundle overM

I G(connected component of 1 of) isometry group of(N,g)

I A= area functional, V= volume functional

I λ= mean curvature (up to a factor)

A set of axioms for equivariant constrained bifurcation — 1

(A1) Msmooth manifold modeled on a separable Banach spaceX;

(A2) Gcompact (connected) Lie group actiongcontinuouslyonM;

(A3) A:M→RsmoothG-invariant function;

(A4) V :M→RsmoothG-invariant function without critical points; (A5) orbits of critical points off_λ=A+λ· Vsmoooth submanifolds of

M.

In the CMC variational problem:

I M=Emb(M,N)/Diff(M)

I X= space of sections of some vector bundle overM

I G(connected component of 1 of) isometry group of(N,g)

I A= area functional, V= volume functional

I λ= mean curvature (up to a factor)

A set of axioms for equivariant constrained bifurcation — 1

(A1) Msmooth manifold modeled on a separable Banach spaceX;

(A2) Gcompact (connected) Lie group actiongcontinuouslyonM;

(A3) A:M→RsmoothG-invariant function;

(A4) V :M→RsmoothG-invariant function without critical points;

(A5) orbits of critical points off_λ=A+λ· Vsmoooth submanifolds of M.

In the CMC variational problem:

I M=Emb(M,N)/Diff(M)

I X= space of sections of some vector bundle overM

I G(connected component of 1 of) isometry group of(N,g)

I A= area functional, V= volume functional

I λ= mean curvature (up to a factor)

A set of axioms for equivariant constrained bifurcation — 1

(A1) Msmooth manifold modeled on a separable Banach spaceX;

(A2) Gcompact (connected) Lie group actiongcontinuouslyonM;

(A3) A:M→RsmoothG-invariant function;

(A4) V :M→RsmoothG-invariant function without critical points;

(A5) orbits of critical points off_λ=A+λ· Vsmoooth submanifolds of M.

In the CMC variational problem:

I M=Emb(M,N)/Diff(M)

I X= space of sections of some vector bundle overM

I G(connected component of 1 of) isometry group of(N,g)

I A= area functional, V= volume functional

I λ= mean curvature (up to a factor)

A set of axioms for equivariant constrained bifurcation — 1

(A1) Msmooth manifold modeled on a separable Banach spaceX;

(A2) Gcompact (connected) Lie group actiongcontinuouslyonM;

(A3) A:M→RsmoothG-invariant function;

(A4) V :M→RsmoothG-invariant function without critical points;

(A5) orbits of critical points off_λ=A+λ· Vsmoooth submanifolds of M.

In the CMC variational problem:

I M=Emb(M,N)/Diff(M)

I X= space of sections of some vector bundle overM

I G(connected component of 1 of) isometry group of(N,g)

I A= area functional, V= volume functional

I λ= mean curvature (up to a factor)

A set of axioms for equivariant constrained bifurcation — 1

(A1) Msmooth manifold modeled on a separable Banach spaceX;

(A2) Gcompact (connected) Lie group actiongcontinuouslyonM;

(A3) A:M→RsmoothG-invariant function;

(A4) V :M→RsmoothG-invariant function without critical points;

(A5) orbits of critical points off_λ=A+λ· Vsmoooth submanifolds of M.

In the CMC variational problem:

I M=Emb(M,N)/Diff(M)

I X= space of sections of some vector bundle overM

I G(connected component of 1 of) isometry group of(N,g)

I A= area functional, V= volume functional

I λ= mean curvature (up to a factor)

A set of axioms for equivariant constrained bifurcation — 1

(A1) Msmooth manifold modeled on a separable Banach spaceX;

(A2) Gcompact (connected) Lie group actiongcontinuouslyonM;

(A3) A:M→RsmoothG-invariant function;

(A4) V :M→RsmoothG-invariant function without critical points;

(A5) orbits of critical points off_λ=A+λ· Vsmoooth submanifolds of M.

In the CMC variational problem:

I M=Emb(M,N)/Diff(M)

I X= space of sections of some vector bundle overM

I G(connected component of 1 of) isometry group of(N,g)

I A= area functional, V= volume functional

I λ= mean curvature (up to a factor)

A set of axioms for equivariant constrained bifurcation — 1

(A1) Msmooth manifold modeled on a separable Banach spaceX;

(A2) Gcompact (connected) Lie group actiongcontinuouslyonM;

(A3) A:M→RsmoothG-invariant function;

(A4) V :M→RsmoothG-invariant function without critical points;

(A5) orbits of critical points off_λ=A+λ· Vsmoooth submanifolds of M.

In the CMC variational problem:

I M=Emb(M,N)/Diff(M)

I X= space of sections of some vector bundle overM

I G(connected component of 1 of) isometry group of(N,g)

I A= area functional, V= volume functional

I λ= mean curvature (up to a factor)

A set of axioms for equivariant constrained bifurcation — 1

(A1) Msmooth manifold modeled on a separable Banach spaceX;

(A2) Gcompact (connected) Lie group actiongcontinuouslyonM;

(A3) A:M→RsmoothG-invariant function;

(A4) V :M→RsmoothG-invariant function without critical points;

(A5) orbits of critical points off_λ=A+λ· Vsmoooth submanifolds of M.

In the CMC variational problem:

I M=Emb(M,N)/Diff(M)

I X= space of sections of some vector bundle overM

I G(connected component of 1 of) isometry group of(N,g)

I A= area functional, V = volume functional

I λ= mean curvature (up to a factor)

A set of axioms for equivariant constrained bifurcation — 1

(A1) Msmooth manifold modeled on a separable Banach spaceX;

(A2) Gcompact (connected) Lie group actiongcontinuouslyonM;

(A3) A:M→RsmoothG-invariant function;

(A4) V :M→RsmoothG-invariant function without critical points;

(A5) orbits of critical points off_λ=A+λ· Vsmoooth submanifolds of M.

In the CMC variational problem:

I M=Emb(M,N)/Diff(M)

I X= space of sections of some vector bundle overM

I G(connected component of 1 of) isometry group of(N,g)

I A= area functional, V = volume functional

I λ= mean curvature (up to a factor)

A set of axioms for equivariant constrained bifurcation — 2 Hilbertization & Fredholmness A

(HF-A) gradient mapforfλ: For allλ₀and allx₀∈Crit(fλ₀),∃U neighborhood ofx0, a Banach spaceY, a Hilbert spaceH0, with continuous dense inclusions:

X,→Y,→H₀,

and a mapF : ]λ₀−ε, λ₀+ε[×U−→Ysuch that:

I dfλ(x)v=

F(λ,x),v

H₀

I ∂F

∂x(λ0,x0) :X−→YFredholm of index 0. (HF-A) implies:

(a) localPalais–Smale conditionforfλ;

(b) manifold structure of critical orbits near nondegenerate ones, via Implicit Function Theorem.

A set of axioms for equivariant constrained bifurcation — 2 Hilbertization & Fredholmness A

(HF-A) gradient mapforfλ: For allλ₀and allx₀∈Crit(fλ₀),∃U neighborhood ofx0, a Banach spaceY, a Hilbert spaceH0, with continuous dense inclusions:

X,→Y,→H0,

and a mapF : ]λ₀−ε, λ₀+ε[×U−→Ysuch that:

I dfλ(x)v=

F(λ,x),v

H₀

I ∂F

∂x(λ0,x0) :X−→YFredholm of index 0.

(HF-A) implies:

(a) localPalais–Smale conditionforfλ;

(b) manifold structure of critical orbits near nondegenerate ones, via Implicit Function Theorem.

A set of axioms for equivariant constrained bifurcation — 2 Hilbertization & Fredholmness A

(HF-A) gradient mapforfλ: For allλ₀and allx₀∈Crit(fλ₀),∃U neighborhood ofx0, a Banach spaceY, a Hilbert spaceH0, with continuous dense inclusions:

X,→Y,→H0,

and a mapF : ]λ₀−ε, λ₀+ε[×U−→Ysuch that:

I dfλ(x)v=

F(λ,x),v

H₀

I ∂F

∂x(λ0,x0) :X−→YFredholm of index 0.

(HF-A) implies:

(a) localPalais–Smale conditionforfλ;

(b) manifold structure of critical orbits near nondegenerate ones, via Implicit Function Theorem.

A set of axioms for equivariant constrained bifurcation — 2 Hilbertization & Fredholmness A

(HF-A) gradient mapforfλ: For allλ₀and allx₀∈Crit(fλ₀),∃U neighborhood ofx0, a Banach spaceY, a Hilbert spaceH0, with continuous dense inclusions:

X,→Y,→H0,

and a mapF : ]λ₀−ε, λ₀+ε[×U−→Ysuch that:

I dfλ(x)v=

F(λ,x),v

H₀

I ∂F

∂x(λ0,x0) :X−→YFredholm of index 0.

(HF-A) implies:

(a) localPalais–Smale conditionforf_λ;

(b) manifold structure of critical orbits near nondegenerate ones, via Implicit Function Theorem.

A set of axioms for equivariant constrained bifurcation — 2 Hilbertization & Fredholmness A

(HF-A) gradient mapforfλ: For allλ₀and allx₀∈Crit(fλ₀),∃U neighborhood ofx0, a Banach spaceY, a Hilbert spaceH0, with continuous dense inclusions:

X,→Y,→H0,

and a mapF : ]λ₀−ε, λ₀+ε[×U−→Ysuch that:

I dfλ(x)v=

F(λ,x),v

H₀

I ∂F

∂x(λ0,x0) :X−→YFredholm of index 0.

(HF-A) implies:

(a) localPalais–Smale conditionforf_λ;

(b) manifold structure of critical orbits near nondegenerate ones, via Implicit Function Theorem.

A set of axioms for equivariant constrained bifurcation — 2 Hilbertization & Fredholmness A

For the CMC problem:

Separability!!!

I X=C^k,α-sections of the normal bundlex₀^⊥,k ≥2,α∈]0,1[;

I Y=C^k−2,α-sections ofx₀^⊥

I H₀=L²-sections ofx₀^⊥

I F quasi-linear 2^nd-order elliptic operator:

div ∇u

p1+k∇uk²

!

I ∂F

∂x Jacobi operator.

Strong ellipticity

Schauder’s estimates

=⇒

Fredholmness

A set of axioms for equivariant constrained bifurcation — 2 Hilbertization & Fredholmness A

For the CMC problem:

Separability!!!

I X=C^k,α-sections of the normal bundlex₀^⊥,k ≥2,α∈]0,1[;

I Y=C^k−2,α-sections ofx₀^⊥

I H₀=L²-sections ofx₀^⊥

I F quasi-linear 2^nd-order elliptic operator:

div ∇u

p1+k∇uk²

!

I ∂F

∂x Jacobi operator.

Strong ellipticity

Schauder’s estimates

=⇒

Fredholmness

A set of axioms for equivariant constrained bifurcation — 2 Hilbertization & Fredholmness A

For the CMC problem:

Separability!!!

I X=C^k,α-sections of the normal bundlex₀^⊥,k ≥2,α∈]0,1[;

I Y=C^k−2,α-sections ofx₀^⊥

I H₀=L²-sections ofx₀^⊥

I F quasi-linear 2^nd-order elliptic operator:

div ∇u

p1+k∇uk²

!

I ∂F

∂x Jacobi operator.

Strong ellipticity

Schauder’s estimates

=⇒

Fredholmness

A set of axioms for equivariant constrained bifurcation — 2 Hilbertization & Fredholmness A

For the CMC problem:

Separability!!!

I X=C^k,α-sections of the normal bundlex₀^⊥,k ≥2,α∈]0,1[;

I Y=C^k−2,α-sections ofx₀^⊥

I H₀=L²-sections ofx₀^⊥

I F quasi-linear 2^nd-order elliptic operator:

div ∇u

p1+k∇uk²

!

I ∂F

∂x Jacobi operator.

Strong ellipticity

Schauder’s estimates

=⇒

Fredholmness

A set of axioms for equivariant constrained bifurcation — 2 Hilbertization & Fredholmness A

For the CMC problem:

Separability!!!

I X=C^k,α-sections of the normal bundlex₀^⊥,k ≥2,α∈]0,1[;

I Y=C^k−2,α-sections ofx₀^⊥

I H₀=L²-sections ofx₀^⊥

I F quasi-linear 2^nd-order elliptic operator:

div ∇u

p1+k∇uk²

!

I ∂F

∂x Jacobi operator.

Strong ellipticity

Schauder’s estimates

=⇒

Fredholmness

A set of axioms for equivariant constrained bifurcation — 2 Hilbertization & Fredholmness A

For the CMC problem:

Separability!!!

I X=C^k,α-sections of the normal bundlex₀^⊥,k ≥2,α∈]0,1[;

I Y=C^k−2,α-sections ofx₀^⊥

I H₀=L²-sections ofx₀^⊥

I F quasi-linear 2^nd-order elliptic operator:

div ∇u

p1+k∇uk²

!

I ∂F

∂x Jacobi operator.

Strong ellipticity

Schauder’s estimates

=⇒

Fredholmness

A set of axioms for equivariant constrained bifurcation — 2 Hilbertization & Fredholmness A

For the CMC problem:

Separability!!!

I X=C^k,α-sections of the normal bundlex₀^⊥,k ≥2,α∈]0,1[;

I Y=C^k−2,α-sections ofx₀^⊥

I H₀=L²-sections ofx₀^⊥

I F quasi-linear 2^nd-order elliptic operator:

div ∇u

p1+k∇uk²

!

I ∂F

∂x Jacobi operator.

Strong ellipticity

Schauder’s estimates

=⇒

Fredholmness

No documento Bifurcation of CMC Clifford Tori in Euclidean Spheres (páginas 44-72)