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

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 $\Lambda$.