2024/02/02 by Battaglia, Luca, Pistoia, Angela, Provenzano, Luigi
#Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Spectral Theory (math.SP)
paper · doi:10.48550/arxiv.2402.01190
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.