We consider the Steklov eigenvalues on a Riemannian manifold Ω with boundary ∂Ω. These are the eigenvalues of the Dirichlet-to-Neumann map. The governing equations of the Steklov problem model the stationary heat distribution inside Ω where the flux through the boundary is proportional to the temperature on the boundary.

We focus our attention on the Steklov eigenvalues on a hypersurface Ω ⊂ Rⁿ⁺¹ , n ≥ 2, with prescribed, smooth boundary ∂Ω ⊂ Rⁿ . We present and discuss an upper bound for these eigenvalues which is independent of the topology and of the curvature of Ω. This is based on an ongoing project with Bruno Colbois (Neuchâtel) and Alexandre Girouard (Laval).