An extremal eigenvalue problem for surfaces with boundary

An extremal eigenvalue problem for surfaces with boundary Richard Schoen Stanford University - Conference in Geometric Analysis, UC Irvine - January 15, 2012 - Joint project with Ailana Fraser

Plan of Lecture The general lecture plan: Part 1: Introduction to Steklov eigenvalues and coarse upper bounds. Part 2: Connections with the minimal surface free boundary value problem. Part 3: Minimal surface uniqueness theorems. Part 4: Main theorems on sharp upper bounds.

Part 1: Introduction, Compact surfaces I For compact Riemannian surfaces M the problem of determining the metric of fixed area which maximizes the first eigenvalue has been solved for surfaces with χ(m) 0: For S 2 the constant curvature metric is the unique maximum by a result of J. Hersch from the 1960s. For RP 2 the constant curvature metric is the unique maximum by a result of P. Li and S. T. Yau from the 1980s. They made the connection with minimal surfaces in spheres; in particular, the Veronese minimal embedding of RP 2 into S 4 is key. For T 2 the flat metric on the 60 0 rhombic torus is the unique maximum by a result of N. Nadirashvili from 1996. It can be minimally embedded into S 5 by first eigenfunctions. In 2000 A. El Soufi and S. Ilias showed that the only other smooth critical metric is the flat square torus which can be minimally embedded into S 3 as the Clifford torus.

Compact surfaces II For the Klein bottle the extremal metric is smooth and unique but not flat. This follows from work of Nadirashvili (1996 existence of maximizer), D. Jacobson, Nadirashvili, and I. Polterovich (2006 constructed the metric), and El Soufi and Ilias (2006 proved it is unique). The metric arises on a minimal immersion of the Klein bottle into S 4. For the genus 2 surface there is a conjectured maximal metric by Jakobson, M. Levitin, Nadirashvili, N. Nigam, Polterovich (2005). It is a singular metric of constant positive curvature with 6 singular points. It arises from a specific complex structure and a branched double cover of S 2. The authors give numerical evidence that the metric maximizes.

Steklov eigenvalues I (M, M) Riemannian manifold Given a function u C ( M), let û be the harmonic extension of u: { g û = 0 on M, û = u on M. The Dirichlet-to-Neumann map is the map given by L : C ( M) C ( M) Lu = û ν. (non-negative, self-adjoint operator with discrete spectrum) Eigenvalues σ 0, σ 1, σ 2,, σ 3,... (Steklov Eigenvalues)

Steklov eigenvalues II Constant fcns are in the kernel of L The lowest eigenvalue of L is zero, σ 0 = 0 The first nonzero eigenvalue σ 1 of L can be characterized variationally as: M σ 1 = û 2 dv M. M u2 dv M inf u C 1 ( M), M u=0 Example: B m, σ k = k, k = 0, 1, 2,... u homogeneous harmonic polynomial of degree k σ 1 = 1 eigenspace x 1,..., x n

An eigenvalue estimate I Weinstock 1954 Ω R 2 simply connected domain σ 1 (Ω)L( Ω) 2π = σ 1 (D)L( D) = only if Ω is a disk Parallel to Szegö s estimate for Neumann eigenvalue µ 1 (Ω)A(Ω) µ 1 (D)A(D) Proof. RMT = proper conformal map of degree 1, ϕ : Ω D conformal F : D D such that (F ϕ) ds = 0 Ω

An eigenvalue estimate II i.e. WLOG, Ω ϕ ds = 0. Then, for i = 1, 2, σ 1 ϕ 2 i ds ˆϕ i 2 da Ω Ω ϕ i 2 da Ω

An eigenvalue estimate II i.e. WLOG, Ω ϕ ds = 0. Then, for i = 1, 2, σ 1 ϕ 2 i ds ˆϕ i 2 da Ω Ω ϕ i 2 da Ω σ 1 Ω 2 i=1 ϕ 2 i ds Ω 2 ϕ i 2 da i=1 σ 1 L( Ω) 2 A(ϕ(Ω)) = 2A(D) = 2π

An eigenvalue estimate II i.e. WLOG, Ω ϕ ds = 0. Then, for i = 1, 2, σ 1 ϕ 2 i ds ˆϕ i 2 da Ω Ω ϕ i 2 da Ω σ 1 Ω 2 i=1 ϕ 2 i ds Ω 2 ϕ i 2 da i=1 σ 1 L( Ω) 2 A(ϕ(Ω)) = 2A(D) = 2π Payne, Hersch, Bandle, Schiffer, Escobar, Girouard-Polterovich...

Spectral equivalence for surfaces We call two surfaces M 1 and M 2 σ-isometric (resp. σ-homothetic) if there is a conformal diffeomorphism F : M 1 M 2 which is an isometry (resp. a homothety) on the boundary; that is, F g 2 = λ 2 g 1 and λ = 1 on M 1 (resp. λ = c on M 1 for some positive constant c). If M 1 and M 2 are σ-isometric then they have the same Steklov eigenvalues. If M 1 and M 2 are σ-homothetic then they have the same normalized Steklov eigenvalues; that is, σ i (M 1 )L 1 = σ i (M 2 )L 2 for each i.

Coarse upper bounds Theorem: (M 2, M) oriented Riemannian surface of genus γ with k boundary components. Then, σ 1 L( M) min{2π(γ + k), 8π[(γ + 3)/2]} The inequality is strict if γ = 0 and k > 1. Note: γ = 0, k = 1 simply connected surface: Weinstock. Thanks to G. Kokarev for pointing out the second bound. Proof: Alhfors and Gabard: deg(ϕ) γ + k. proper conformal ϕ : M D with Fill boundary components with disks to get a closed surface M of genus γ and use the result that a holomorphic map to S 2 with degree at most [(γ + 3)/2]. Use Szegö/Hersch trick of balancing map as before.

Rotationally symmetric metrics on the annulus [ T, T ] S 1 for some T > 0 g = f 2 (t)(dt 2 + dθ 2 ) σ 1 (T )L(T ) σ 1 (T 0 )L(T 0 ) Max achieved when L 1 = L 2

Critical Catenoid For T = T 0, f (t) = 1 eigenfunctions are emb. fcns for catenoid x = cosh t cos θ y = cosh t sin θ z = t

Critical Catenoid For T = T 0, f (t) = 1 eigenfunctions are emb. fcns for catenoid x = cosh t cos θ y = cosh t sin θ z = t Critical catenoid is embedded by 1st Steklov e.f. and maximizes σ 1 L over all rotationally sym. metrics on annuli.

Part 2. Minimal submanifolds in the ball (M k, M k ) (B n, B n ) M minimal, meeting B n orthogonally along M H = 0 η = x M R n minimal M x i = 0 i = 1,..., n (x 1,... x n are harmonic) M meets B n orthogonally x i η = x i, i = 1,..., n σ = 1 is a Steklov eigenvalue with x 1,..., x n eigenfunctions

Examples I 1) M k = D k B n equatorial k-plane J.C.C. Nitsche: M 2 simply connected in B 3 = M flat disk.

Examples I 1) M k = D k B n equatorial k-plane J.C.C. Nitsche: M 2 simply connected in B 3 = M flat disk 2) Critical Catenoid

Examples II 3) Critical Möbius Band We think of the Möbius band M as R S 1 with the identification (t, θ) ( t, θ + π). There is a minimal embedding of M into R 4 given by ϕ(t, θ) = (2 sinh t cos θ, 2 sinh t sin θ, cosh 2t cos 2θ, cosh 2t sin 2θ) For a unique choice of T 0 the restriction of ϕ to [ T 0, T 0 ] S 1 defines an embedding into a ball by first Steklov eigenfunctions. We may rescale the radius of the ball to 1 to get the critical Möbius band. Explicitly T 0 is the unique positive solution of coth t = 2 tanh 2t.

Examples III 4) Minimal cones M k 1 minimal in S n 1 C(M) cone over M Variational Theory: Almgren, Gromov, L. Guth, Fraser ( th, index estimates) Remark: k M = M x 2 = 2k

Part 2: Minimal surface uniqueness theorems In order to show that the critical catenoid achieves the maximum of σ 1 L we must characterize it among the free boundary minimal annuli. Note that the theorem of Nitsche together with a multiplicity bound implies that the flat disk is the only free boundary minimal disk in B n immersed by first Steklov eigenfunctions. The next results extend this to the annulus and Möbius band. Theorem A: Assume that Σ is a free boundary minimal annulus in B n such that the coordinate functions are first eigenfunctions. Then n = 3 and Σ is the critical catenoid.

Theorem A: Proof outline I A multiplicity bound implies that n = 3. We may assume that Σ is parametrized by a conformal map ϕ from M = [ T, T ] S 1 with coordinates (t, θ). The vector field X = ϕ θ is then a conformal Killing vector field along Σ. The goal is to show that X coincides with a rotation vector field of R 3. The key step in doing this is to show that the three components of X are first eigenfunctions, and therefore are linear combinations of the coordinate functions. For functions or vector fields Y defined along Σ we consider the quadratic form Q defined by Q(Y, Y ) = Y 2 da Y 2 ds. Σ We observe that for Y which integrate to 0 on Σ we have Q(Y, Y ) 0 with equality if and only if the components of Y are linear functions. This follows from the assumption that the first eigenvalue of Σ is 1. Σ

Proof outline II The quadratic form Q is also the second variation of 1 2 E provided that Y is tangent to S 2 along Σ. It is easy to check that the vector field X is in the nullspace of Q. If it were true that X ds = 0, then we could complete the argument. Σ The idea is to find a vector field Y such that Q(Y, Y ) 0 and with Σ (X Y ) ds = 0. It would then follow that Q(X Y, X Y ) 0, and it would follow that the components of X Y are linear functions. There is an obvious choice to make for Y, namely a conformal Killing vector field in the ball. This amounts to taking a vector v R 3 and setting Y (v) = 1 + x 2 v (x v)x 2 where x is the position vector in R 3. It is easy to check that v can be chosen so that Σ (X Y ) ds = 0.

Proof outline III Since Y is conformal, Q(Y, Y ) is the second variation of the area functional. If it is true that Q(Y, Y ) 0, we could complete the proof. Unfortunately we were not able to show this despite much effort. To get around this difficulty we considered the second variation of area for normal variations. Note that for free boundary solutions there is a natural Jacobi field given by x ν where ν is the unit normal vector of Σ. It is in the nullspace of the second variation form S for area given by S(ψ, ψ) = ( ψ 2 A 2 ψ 2 ) da ψ 2 ds Σ where A denotes the second fundamental form of Σ and we are considering the normal variation ψν. Note that this variation is tangent to S 2 along the boundary. Σ

Proof outline IV We can show by a subtle argument that for any v R 3 we have S(v ν, v ν) 0. This is not sufficient for the eigenvalue problem because the normal deformation does not preserve the conformal structure of Σ in general. The way we get around this problem is to consider adding a tangential vector field Y t to so that Y = Y t + ψν preserves the conformal structure and is tangent to S 2 along Σ. This involves solving a Cauchy-Riemann equation with boundary condition to determine Y t. This problem is generally not solvable, but has a 1 dimensional obstruction for its solvability (because Σ is an annulus). We then get existence for ψ in a three dimensional subspace of the span of ν 1, ν 2, ν 3, x ν. We can then arrange the resulting conformal vector field Y to satisfy the boundary integral condition and we have Q(Y, Y ) = S(ψ, ψ) = S(v ν, v ν) 0.

Theorem B: Proof outline A multiplicity bound implies that n 5. The case n = 2 cannot occur since there is no proper conformal map from a Möbius band onto the disk. The previous argument can be modified to show that the surface Σ is invariant under a one parameter group of rotations of R n. For n = 3 and n = 5 this group would preserve an axis and there would be a coordinate function which is independent of θ which is a contradiction. It follows that n = 4 and one can check that the surface is congruent to the critical Möbius band.

Part 4: Main theorems on sharp bounds Let σ (γ, k) denote the supremum of σ 1 L over metrics on a surface of genus γ with k boundary components. We know σ (0, 1) = 2π. The next result identifies σ (0, 2). Theorem 1: For any metric annulus M we have σ 1 L (σ 1 L) cc with equality iff M is σ-homothetic to the critical catenoid. In particular, σ (0, 2) = (σ 1 L) cc 4π/1.2. Theorem 2: The sequence σ (0, k) is strictly increasing in k and converges to 4π as k tends to infinity. For each k a maximizing metric is achieved by a free boundary minimal surface Σ k in B 3 of area less than 2π. The limit of these minimal surfaces as k tends to infinity is a double disk, and for large k, Σ k is approximately a pair of nearby parallel plane disks joined by k boundary bridges.

Main theorems II Corollary: For every k 1 there is an embedded minimal surface in B 3 of genus 0 with k boundary components satisfying the free boundary condition. Moreover these surfaces are embedded by first eigenfunctions. Here is a rough sketch of the surfaces for large k.

Maximizing metrics The following result shows that a metric which maximizes σ 1 L arises from a free boundary minimal surface in a ball. Theorem: Let M be a compact surface with nonempty boundary and assume that g is a metric on M for which σ 1 L is maximized. The multiplicity of σ 1 is at least two, and there exists a proper branched minimal immersion ϕ : M B n for some n 2 by first eigenfunctions which is a homothety on the boundary. The surface Σ = ϕ(m) is a free boundary minimal surface.

Proof outline I The proof involves a modification of an idea of Nadirashvili for closed surfaces. Step 1: Compute the first variation of the eigenvalue σ(t) for a smoothly varying path of eigenfunctions u t for a family of metrics g t with fixed boundary length. This may be computed most easily by differentiating at t = 0 u t 2 t da t = σ(t) ut 2 ds t. M where g t is a path of metrics with g 0 = g. If we denote derivatives at t = 0 with dots we let h = ġ, and the length constraint on the boundary translates to h(t, T ) ds = 0 M where T denotes the oriented unit tangent vector to M. M

Proof outline II Denoting the derivative of σ at t = 0 by σ, σ = τ(u), h da u 2 h(t, T ) ds M M where τ(u) is the stress-energy tensor of u given by τ(u) ij = i u j u 1 2 u 2 g ij

Proof outline II Denoting the derivative of σ at t = 0 by σ, σ = τ(u), h da u 2 h(t, T ) ds M M where τ(u) is the stress-energy tensor of u given by τ(u) ij = i u j u 1 2 u 2 g ij Step 2: Assuming that σ 1 L is maximized for g, show that for any variation h there exists an eigenfunction u (depending on h) with Q h (u) := τ(u), h da + u 2 h(t, T ) ds = 0. M M

This is accomplished by using left and right hand derivatives to show that the quadratic form Q h is indefinite and therefore has a null vector. Proof outline II Denoting the derivative of σ at t = 0 by σ, σ = τ(u), h da u 2 h(t, T ) ds M M where τ(u) is the stress-energy tensor of u given by τ(u) ij = i u j u 1 2 u 2 g ij Step 2: Assuming that σ 1 L is maximized for g, show that for any variation h there exists an eigenfunction u (depending on h) with Q h (u) := τ(u), h da + u 2 h(t, T ) ds = 0. M M

Proof outline III Step 3: Use the Hahn-Banch theorem in an appropriate Hilbert space consisting of pairs (p, f ) where p is a symmetric (0, 2) tensor on M and f a function on M to show that the pair (0, 1) lies in the convex hull of the pairs (τ(u), u 2 ) for first eigenfunctions u. This tells us that there are positive a 1,..., a n and eigenfunctions u 1,..., u n so that a 2 i τ(u i ) = 0 on M, a 2 i ui 2 = 1 on M. It then follows that the map ϕ = (a 1 u 1,..., a n u n ) is a (possibly branched) proper minimal immersion in the unit ball B n. It can then be checked that ϕ is a homothety on M.

Existence of maximizers Theorem: For any k 1 there is a smooth metric on the surface of genus 0 with k boundary components with the property σ 1 L = σ (0, k).

Existence of maximizers Theorem: For any k 1 there is a smooth metric on the surface of genus 0 with k boundary components with the property σ 1 L = σ (0, k). The first step is to show that σ (0, k) > σ (0, k 1) and that the conformal structure does not degenerate for a maximizing sequence. Next we take a maximizing sequence and a weak* limit of the boundary measures. We then prove that first eigenfunctions give a branched conformal minimal immersion into the ball which is freely stationary. The regularity at the boundary then follows.

The limit as k goes to infinity I Let Σ k B 3 be a maximizing surface with genus 0 and k boundary components. We have the following properties.

The limit as k goes to infinity I Let Σ k B 3 be a maximizing surface with genus 0 and k boundary components. We have the following properties. Σ k does not contain the origin and is embedded and star shaped. This follows from the fact that the restrictions of the linear functions have no critical points on their zero set. The coarse upper bound implies that A(Σ k ) 4π and the star shaped property implies that each Σ k is stable for variations which fix the boundary. Curvature estimates then imply uniform curvature bounds in the interior.

The limit as k goes to infinity II There is a subsequence of the Σ k which converges in a smooth topology to a smooth limiting minimal surface Σ possibly with multiplicity. This limit must have multiplicity since otherwise the limit would be a smooth free boundary solution, and the convergence would be smooth up to the boundary contradicting the fact that k. It follows from the star shaped condition that the origin lies in the limiting surface, the surface is a cone (hence a flat disk since it is smooth), and the multiplicity is 2.

The Möbius band Finally we show that the critical Möbius band uniquely maximizes σ 1 L. After some calculation one can see that (σ 1 L) cmb = 6 6π. We have the result. Theorem 1: For any metric on the Möbius band M we have σ 1 L (σ 1 L) cmb with equality iff M is σ-homothetic to the critical Möbius band. The proof follows the same steps as for the critical catenoid. We show the existence of a smooth maximizing metric on the Möbius band. This then gives an immersion into B n by first eigenfunctions, and by the uniqueness result, this immersion is congruent to the critical Möbius band

Part 5: Volumes of Minimal Submanifolds We have the following result on conformal images of free boundary surfaces. Proposition: Suppose f : B n B n is conformal. Then L(f ( Σ)) L( Σ)

Part 5: Volumes of Minimal Submanifolds We have the following result on conformal images of free boundary surfaces. Proposition: Suppose f : B n B n is conformal. Then L(f ( Σ)) L( Σ) Proof. We use a first variation argument. Recall that the first variation formula is div Σ (V ) = V x ds Σ Σ where div Σ (V ) = i e i V e i in an orthonormal tangent basis.

Completion of proof Let u be the length magnification factor of f ; that is, f (δ) = u 2 δ where δ is the Euclidean metric. It can be checked that there is a y R n with y > 1 so that u(x) = y 2 1 x y 2. If we make the choice V = x y x y 2 in the first variation we can check that div Σ V 0 and V x = 1 2 (1 u) on Σ. It then follows that 0 (1 u) ds = Σ f ( Σ). Σ Remark: The proof above uses only the first variation, and thus works for integer multiplicity rectifiable varifolds which are stationary for deformations which preserve the ball. This is the class of objects which are produced by Almgren in the min/max theory for the variational problem.

Lower area bounds for free boundary surfaces A lower bound on areas follows from the boundary length decrease under conformal maps of the ball since such maps can be chosen which blow up any chosen point x Σ and the limit is an equatorial disk with area π. Theorem: Let Σ 2 be a minimal surface in B n with Σ B n, meeting B n orthogonally. Then Σ π ( equivalently Σ 2π since Σ = 2 Σ ) Note: Σ 2π Σ Σ 2 4π ( Σ 2 Σ 2 4π ) sharp isoperimetric inequality

Higher dimensional volume bounds Conjecture: If Σ is a k dimensional minimal submanifold in the ball B n with Σ S n 1 and with the conormal vector of Σ equal to the position vector, then for any conformal transformation f of the ball we have f ( Σ) Σ. Evidence: It is true for k = 2, and in the special case of a cone, results of Li and Yau for k = 3 and of El Soufi and Ilias for k 4 imply the conjecture. It is true in general that the boundary volume is reduced up to second order. A consequence of the conjecture is that Σ B k. This inequality was shown recently by a direct monotonicity-style argument by S. Brendle.