One can hear the area and curvature of boundary of a domain by hearing the Steklov eigenvalues

TL;DR

This paper establishes a connection between the Steklov eigenvalues and geometric properties like area and curvature of a domain's boundary by analyzing the heat kernel associated with the Dirichlet-to-Neumann operator.

Contribution

It develops a method to compute the asymptotic expansion coefficients of the heat kernel trace, linking spectral data to boundary geometry.

Findings

Explicit formulas for the first four coefficients of the heat kernel expansion.

Demonstrates equivalence of various inequalities related to the Dirichlet-to-Neumann operator.

Provides a procedure to recover boundary curvature and area from Steklov spectrum.

Abstract

For a given bounded domain $\Omega$ with smooth boundary in a smooth Riemannian manifold $(\mathcal{M},g)$, we show that the Poisson type upper-estimate of the heat kernel associated to the Dirichlet-to-Neumann operator, the Sobolev trace inequality, the Log-Sobolev trace inequality, the Nash trace inequality, and the Rozenblum-Lieb-Cwikel type inequality are all equivalent. Upon decomposing the Dirichlet-to-Neumann operator into a sum of the square root of the Laplacian and a pseudodifferntial operator and by applying Grubb's method of symbolic calculus for the corresponding pseudodifferential heat kernel operators, we establish a procedure to calculate all the coefficients of the asymptotic expansion of the trace of the heat kernel associated to Dirichlet-to-Neumann operator as $t\to 0^+$. In particular, we explicitly give the first four coefficients of this asymptotic expansion. These coefficients give precise information regarding the area and curvatures of the boundary of the domain in terms of the spectrum of the Steklov problem.