On the critical points of Steklov eigenfunctions

We consider the critical points of Steklov eigenfunctions on a compact, smooth n-dimensional Riemannian manifold M with boundary ∂M. For generic metrics on M we establish an identity which relates the sum of the indexes of a Steklov eigenfunction, the sum of the indexes of its restriction to ∂M, and the Euler characteristic of M. In dimension 2 this identity gives a precise count of the interior critical points of a Steklov eigenfunction in terms of the Euler characteristic of M and of the number of sign changes of u on ∂M. In the case of the second Steklov eigenfunction on a genus 0 surface, the identity holds for any metric. As a by-product of the main result, we show that for generic metrics on M Steklov eigenfunctions are Morse functions in M.