# Some results on higher eigenvalue optimization

@article{Fraser2020SomeRO, title={Some results on higher eigenvalue optimization}, author={Ailana M. Fraser and Richard M. Schoen}, journal={Calculus of Variations and Partial Differential Equations}, year={2020} }

In this paper we obtain several results concerning the optimization of higher Steklov eigenvalues both in two and higher dimensional cases. We first show that the normalized (by boundary length) $k$-th Steklov eigenvalue on the disk is not maximized for a smooth metric on the disk for $k\geq 3$. For $k=1$ the classical result of [W] shows that $\sigma_1$ is maximized by the standard metric on the round disk. For $k=2$ it was shown [GP1] that $\sigma_2$ is not maximized for a smooth metric. We…

## 13 Citations

Higher Dimensional Surgery and Steklov Eigenvalues

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We show that for compact Riemannian manifolds of dimension at least $3$ with nonempty boundary, we can modify the manifold by performing surgeries of codimension $2$ or higher, while keeping the…

SP ] 1 3 Ju l 2 02 0 HIGHER DIMENSIONAL SURGERY AND STEKLOV EIGENVALUES

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Abstract. We show that for compact Riemannian manifolds of dimension at least 3 with nonempty boundary, we can modify the manifold by performing surgeries of codimension 2 or higher, while keeping…

Extremal Eigenvalue Problems and Free Boundary Minimal Surfaces in the Ball

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The main theme of this chapter is the study of extremal eigenvalue problems and its relations to minimal surface theory. We describe joint work with R. Schoen on progress that has been made on the…

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For any compact surface $\Sigma$ with smooth, non-empty boundary, we construct a free boundary minimal immersion into a Euclidean Ball $\mathbb{B}^N$ where $N$ is controlled in terms of the topology…

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In this work, we focus on three problems. First, we give a relationship between the number of eigenvalues of the Jacobi operator below a certain threshold and the topology of closed constant mean…

Computation of free boundary minimal surfaces via extremal Steklov eigenvalue problems

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Recently Fraser and Schoen showed that the solution of a certain extremal Steklov eigenvalue problem on a compact surface with boundary can be used to generate a free boundary minimal surface, i.e.,…

Metric upper bounds for Steklov and Laplace eigenvalues

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- 2021

We prove two upper bounds for the Steklov eigenvalues of a compact Riemannian manifold with boundary. The first involves the volume of the manifold and of its boundary, as well as packing and volume…

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- 2022

We adopt the convention that each eigenvalue is repeated according to its multiplicity. An important issue in spectral geometry is to obtain good estimates for these and other eigenvalues in terms of…

Existence and Classification of $$\pmb {\mathbb {S}}^1$$-Invariant Free Boundary Minimal Annuli and Möbius Bands in $$\pmb {\mathbb {B}}^n$$

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We explicitly classify all $$\mathbb {S}^1$$-invariant free boundary minimal annuli and Mobius bands in $${\mathbb {B}}^n$$. This classification is obtained from an analysis of the spectrum of the…

Generic properties of Steklov eigenfunctions

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- 2022

A bstract . Let M n be a smooth compact manifolds with smooth boundary. We show that for a generic C k metic on M n with k > n − 1, the nonzero Steklov eigenvalues are simple. Moreover, we also prove…

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