Bifurcation and symmetry breaking in geometric variational problems

Transcription

1 in geometric variational problems Joint work with M. Koiso and B. Palmer Departamento de Matemática Instituto de Matemática e Estatística Universidade de São Paulo EIMAN I ENCONTRO INTERNACIONAL DE MATEMÁTICA NO NORDESTE BRASILEIRO

2 Generalities on variational bifurcation General bifurcation setup: M differentiable manifold (possibly dim = )

3 Generalities on variational bifurcation General bifurcation setup: M differentiable manifold (possibly dim = ) f λ : M R family of smooth functionals, λ [a, b]

4 Generalities on variational bifurcation General bifurcation setup: M differentiable manifold (possibly dim = ) f λ : M R family of smooth functionals, λ [a, b] λ x λ M smooth curve of critical points: df λ (x λ ) = 0 for all λ.

5 Generalities on variational bifurcation General bifurcation setup: Definition M differentiable manifold (possibly dim = ) f λ : M R family of smooth functionals, λ [a, b] λ x λ M smooth curve of critical points: df λ (x λ ) = 0 for all λ. Bifurcation at λ 0 ]a, b[ if λ n λ 0 and x n x λ0 as n, with: (a) df λn (x n ) = 0 for all n; (b) x n x λn for all n.

6 Generalities on variational bifurcation General bifurcation setup: Definition M differentiable manifold (possibly dim = ) f λ : M R family of smooth functionals, λ [a, b] λ x λ M smooth curve of critical points: df λ (x λ ) = 0 for all λ. Bifurcation at λ 0 ]a, b[ if λ n λ 0 and x n x λ0 as n, with: (a) df λn (x n ) = 0 for all n; (b) x n x λn for all n.

7 Equivariant bifurcation Assume: G Lie group acting on M f λ is G-invariant for all λ Note: the orbit G x λ consists of critical points.

8 Equivariant bifurcation Assume: G Lie group acting on M f λ is G-invariant for all λ Note: the orbit G x λ consists of critical points. Definition Orbit bifurcation at λ 0 ]a, b[ if λ n λ 0 and x n x λ0 as n, with: (a) df λn (x n ) = 0 for all n; (b) G x n G x λn for all n.

9 Equivariant bifurcation Assume: G Lie group acting on M f λ is G-invariant for all λ Note: the orbit G x λ consists of critical points. Definition Orbit bifurcation at λ 0 ]a, b[ if λ n λ 0 and x n x λ0 as n, with: (a) df λn (x n ) = 0 for all n; (b) G x n G x λn for all n. Standard bifurcation theory requires a quite involved variational setup: differentiability, Palais Smale, Fredholmness...

10 Equivariant bifurcation Assume: G Lie group acting on M f λ is G-invariant for all λ Note: the orbit G x λ consists of critical points. Definition Orbit bifurcation at λ 0 ]a, b[ if λ n λ 0 and x n x λ0 as n, with: (a) df λn (x n ) = 0 for all n; (b) G x n G x λn for all n. Standard bifurcation theory requires a quite involved variational setup: differentiability, Palais Smale, Fredholmness... Bifurcation occurs at degenerate critical points with jumps of the Morse index. In the equivariant case, bifurcation occurs at degenerate critical orbits where jumps of the critical groups.

11 Constant mean curvature variational problem M m compact oriented manifold

12 Constant mean curvature variational problem M m compact oriented manifold (N n, g) oriented Riemannian manifold

13 Constant mean curvature variational problem M m compact oriented manifold (N n, g) oriented Riemannian manifold n = m + 1

14 Constant mean curvature variational problem M m compact oriented manifold (N n, g) oriented Riemannian manifold n = m + 1 x : M N embedding

15 Constant mean curvature variational problem M m compact oriented manifold (N n, g) oriented Riemannian manifold n = m + 1 x : M N embedding Mean curvature: H x = tr ( 2 nd fund. form )

16 Constant mean curvature variational problem M m compact oriented manifold (N n, g) oriented Riemannian manifold n = m + 1 x : M N embedding Mean curvature: H x = tr ( 2 nd fund. form ) Variational principle x has constant mean curvature (CMC) iff x is a stationary point for the area functional restricted to embeddings of fixed volume.

17 The nodary The nodary and the symmetry axis.

18 The nodary The nodary and the symmetry axis. A portion of nodoid, with boundary on parallel planes orthogonal to the axis.

19 Fixed boundary problem Circles for the CMC fixed boundary problem, lying on the planes Π 0 and Π 1. In the middle, the symmetry plane Π.

20 The 1-parameter family of nodoids Σ a,h,t0 surface of revolution around the x 3 -axis with generatrix the nodary: x 1 (t) = cos t + cos 2 t + a, t [ t 0, t 0 ] 2 H x 3 (t) = 1 t cos τ + cos 2 τ + a cos τ dτ 2 H 0 cos 2 τ + a

21 The 1-parameter family of nodoids Σ a,h,t0 surface of revolution around the x 3 -axis with generatrix the nodary: x 1 (t) = cos t + cos 2 t + a, t [ t 0, t 0 ] 2 H x 3 (t) = 1 t cos τ + cos 2 τ + a cos τ dτ 2 H 0 cos 2 τ + a H= mean curvature a = 2cH from conservation law: 2x 1 cos t + 2Hx 2 1 = c

22 The 1-parameter family of nodoids Σ a,h,t0 surface of revolution around the x 3 -axis with generatrix the nodary: x 1 (t) = cos t + cos 2 t + a, t [ t 0, t 0 ] 2 H x 3 (t) = 1 t cos τ + cos 2 τ + a cos τ dτ 2 H 0 cos 2 τ + a H= mean curvature a = 2cH from conservation law: 2x 1 cos t + 2Hx 2 1 = c proposition There exist real-analytic functions a = a(t 0 ) and H = H(t 0 ) such that Σ t0 = Σ a(t0 ),H(t 0 ),t 0 satisfies the boundary condition.

23 a = a(t 0 )

24 Nodaries through two circles h 0 2 h 0 2 r 0 0 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 when a 0.

25 The Jacobi operator Jf = f (k1 2 + k 2 2 )f Eigenvalues λ 1 < λ 2 <... + Courant s nodal domain theorem Jf = λ k f = f has at least k nodal domains Separation of variables: f = T (θ) S(s) T + κt = 0, T (0) = T (2π), T (0) = T (2π), ( κ ) (xs ) + x x(k k 2 2 ) S = λxs, S ( L ( 2) = S L 2) = 0.

26 Spherical caps with the same boundary

27 Degenerate nodoids Degenerate nodoids are tangent to the planes containing their boundary. On the left, nodoids from the family Σ, on the right nodoids that are not symmetric with respect to the reflection around the plane Π.

28 Degenerate nodoid with one bulge

29 Two nodal domains

30 Six nodal domains

31 The degenerate nodoid Σ π

32 Bifurcating branch of nodoids at the instant t 0 = π

33 Position vector 1

34 A picture of Miyuki and Bennett back