One can know the area and total curvatures of the boundary by hearing the resonances of a Stokes flow

TL;DR

This paper demonstrates that the Steklov eigenvalues of the Dirichlet-to-Neumann map for Stokes flow encode detailed geometric information of the boundary, including area and total mean curvature, through asymptotic spectral analysis.

Contribution

It provides an explicit method to compute all coefficients of the heat kernel's asymptotic expansion, linking spectral data to boundary geometry in Stokes flow.

Findings

Steklov spectral invariants encode boundary area and mean curvature.

Explicit calculation procedure for asymptotic expansion coefficients.

Asymptotic expansion relates spectral data to geometric boundary features.

Abstract

By calculating full symbol for the Dirichlet-to-Neumann map $\Lambda$ of a Stokes flow, we establish the asymptotic expansion of the trace of the heat kernel for $\Lambda$. We also give a useful procedure, by which all coefficients of the asymptotic expansion can be explicitly calculated. These coefficients are the Steklov spectral invariants of $\Lambda$, which provide precise geometric information of the boundary for the Stokes flow. In particular, the first two coefficients show that the area and (total) mean curvature of the the boundary can be known by the Steklov eigenvalues of $\Lambda$.