Small spheres with prescribed nonconstant mean curvature in Riemannian manifolds

TL;DR

This paper proves the existence of a foliation of spheres with prescribed nonconstant mean curvature near non-degenerate critical points of a function on a Riemannian manifold, extending to more general cases with relaxed assumptions.

Contribution

It establishes the existence and essential uniqueness of such foliations around critical points, generalizing previous results to broader conditions.

Findings

Existence of spheres with prescribed mean curvature near critical points

Foliation structure is essentially unique under certain conditions

Relaxation of nondegeneracy still yields meaningful geometric structures

Abstract

Given a function $f$ on a smooth Riemannian manifold without boundary, we prove that if $p \in M$ 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.