Minimal surfaces and eigenvalue problems

By A. Fraser, R. Schoen. 2013

84 citations recorded by the source corpus.

We show that metrics that maximize the k-th Steklov eigenvalue on surfaces with boundary arise from free boundary minimal surfaces in the unit ball. We prove several properties of the volumes of these minimal submanifolds. For free boundary minimal submanifolds in the ball we show that the boundary volume is reduced up to second order under conformal transformations of the ball. For two-dimensional stationary integer multiplicity rectifiable varifolds that are stationary for deformations that preserve the ball, we prove that the boundary length is reduced under conformal transformations. We also give an overview of some of the known results on extremal metrics of the Laplacian on closed surfaces, and give a survey of our recent results from [FS2] on extremal metrics for Steklov eigenvalues on compact surfaces with boundary.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/9ce59805f62daed80ef8eb7acab1b49083705d28

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In Applied Mathematics

In Computational Theory and Mathematics

Citations in the corpus

On a Fractional Ginzburg–Landau Equation and 1/2-Harmonic Maps into Spheres (2013, methodology): “Using the harmonicity of ve and the divergence theorem, it is routine to check that the integral in (2.14) does not depend on the choice of the extension Φ, and that ∂νv e coincides with the classical exterior normal derivative of ve on ω ⊆ ∂Rn+1+ whenever v is smooth.”
Existence and regularity of maximal metrics for the first Laplace eigenvalue on surfaces (2013, unknown): “A lower bound for Λ1 (γ) can be obtained from [3] and [5] (see [15]) : Λ1 (γ) ≥ 3π 4 (γ − 1) .”
Free boundary minimal surfaces of unbounded genus (2016, background): “Fraser-Schoen conjectured (Section 1 in [FS2]) that there should be a sequence of free boundary minimal surfaces with genus approaching infinity converging to the union of the critical catenoid and the disk.”
SPECTRAL GEOMETRY OF THE STEKLOV PROBLEM (2014, unknown): “This result was extended to higher eigenvalues σk in [40].”