Small spheres with prescribed nonconstant mean curvature in Riemannian manifolds

Abstract

Given a function f on a smooth Riemannian manifold without boundary, we prove that if p∈M is a non-degenerate critical point of f, then a neighborhood of p contains a foliation by spheres with mean curvature proportional to f. This foliation is essentially unique. The nondegeneracy assumption can be substantially relaxed, at the expense of losing the property that the family of spheres with prescribed mean curvature defines a foliation.