Upper bounds for Steklov eigenvalues of submanifolds in Euclidean space via the intersection index

TL;DR

This paper establishes new upper bounds for Steklov eigenvalues of submanifolds in Euclidean space, linking geometric intersection indices with eigenvalue estimates, and achieving optimal asymptotic behavior.

Contribution

It introduces explicit upper bounds for Steklov eigenvalues based on intersection indices, volume, and injectivity radius, improving understanding of eigenvalue growth.

Findings

Derived explicit upper bounds involving intersection indices and volumes.

Achieved bounds with optimal asymptotic exponent as eigenvalue index grows.

Connected geometric properties to spectral estimates for submanifolds.

Abstract

We obtain upper bounds for the Steklov eigenvalues $\sigma_k(M)$ of a smooth, compact, connected, $n$-dimensional submanifold $M$ of Euclidean space with boundary $\Sigma$ that involve the intersection indices of $M$ and of $\Sigma$. One of our main results is an explicit upper bound in terms of the intersection index of $\Sigma$, the volume of $\Sigma$ and the volume of $M$ as well as dimensional constants. By also taking the injectivity radius of $\Sigma$ into account, we obtain an upper bound that has the optimal exponent of $k$ with respect to the asymptotics of the Steklov eigenvalues as $k \to \infty$.