The Convexity and the Gaussian Curvature Estimates for the Level Sets of Harmonic Functions on Convex Rings in Space Forms

Abstract

In this paper, we first establish a constant rank theorem for the second fundamental form of the convex level sets of harmonic functions in space forms. Applying the deformation process, we prove that the level sets of the harmonic functions on convex rings in space forms are strictly convex. Moreover, we give a lower bound for the Gaussian curvature of the convex level sets of harmonic functions in terms of the Gaussian curvature of the boundary and the norm of the gradient on the boundary.