Rigidity of a trace estimate for Steklov eigenvalues

Abstract

In this short note, we show the rigidity of a trace estimate for Steklov eigenvalues with respect to functions in our previous work (Shi and Yu (2016) [13]). Namely, we show that equality of the estimate holds if and only if the manifold is a direct product of a round ball and a closed manifold. The key ingredient in the proof is a splitting theorem for flat and totally geodesic Riemannian submersions which may be of independent interests.