The geometry of the first non-zero Stekloff eigenvalue (Q1376567)

Let \(M\) be a compact Riemannian manifold with boundary. Let \(\partial_N\) be the normal derivative with respect to the outward unit normal vector on the boundary. The author considers the solution to the inhomogeous boundary problem \(\Delta\phi=0\) on \(M\) and \(\partial_N\phi=\mu\phi\) on the boundary of \(M\); this is called the Stekloff problem. Let \(\mu_1\) be the first non-trivial eigenvalue of this problem. The solution \(\phi\) represents the steady state temperature on a domain \(M\) where the flux on the boundary is proportional to the temperature; the eigenvalues of the Stekloff problem are the same as the set of eigenvalues for the Dirichlet-Neumann map. The author studies relationships between \(\nu_1\) and the first eigenvalue of the Laplacian on the boundary \(\partial M\) and studies relationships between \(\nu_1\) and the curvatures involved.