Journal of the
Korean Mathematical Society JKMS

Let $(M,g)$ be a compact $m$ dimensional Einstein manifold with smooth boundary. Let $\Delta_{p,\mathcal B}$ be the realization of the $p$ form valued Laplacian with a suitable boundary condition $\mathcal B$. Let ${\rm Spec}(\Delta_{p,\mathcal B})$ be the spectrum where each eigenvalue is repeated according to multiplicity. We show that certain geometric properties of the boundary may be spectrally characterized in terms of this data where we fix the Einstein constant.

Keywords: totally umbillic boundary, totally geodesic boundary, minimal boundary, absolute boundary conditions, relative boundary conditions, Dirichlet Laplacian, Neumann Laplacian

MSC numbers: 58J50