Heat invariants of the Steklov problem

TL;DR

This paper investigates the heat trace asymptotics of the Steklov eigenvalue problem on Riemannian manifolds, explicitly computes initial heat invariants, and demonstrates spectral uniqueness of the 3D ball among Euclidean domains.

Contribution

It describes the structure of Steklov heat invariants, computes the first few explicitly, and proves spectral uniqueness of the 3D ball in Euclidean space.

Findings

Explicit formulas for initial Steklov heat invariants.

The Steklov spectrum uniquely determines a 3D ball among Euclidean domains.

Application of Seeley calculus to Dirichlet-to-Neumann operator.

Abstract

We study the heat trace asymptotics associated with the Steklov eigenvalue problem on a Riemannian manifold with boundary. In particular, we describe the structure of the Steklov heat invariants and compute the first few of them explicitly in terms of the scalar and mean curvatures. This is done by applying the Seeley calculus to the Dirichlet-to-Neumann operator, whose spectrum coincides with the Steklov eigenvalues. As an application, it is proved that a three--dimensional ball is uniquely defined by its Steklov spectrum among all Euclidean domains with smooth connected boundary.