Existence of Self-Cheeger Sets on Riemannian Manifolds

TL;DR

This paper proves the existence of self-Cheeger sets on compact Riemannian manifolds, showing they are perturbations of geodesic balls and form smooth foliations near critical points of scalar curvature.

Contribution

It establishes the existence of self-Cheeger sets as perturbations of geodesic balls, providing new insights into geometric analysis on Riemannian manifolds.

Findings

Existence of self-Cheeger sets near non-degenerate scalar curvature critical points.

Construction of a family of domains perturbing geodesic balls.

Smooth foliation of neighborhoods around critical points.

Abstract

Let $(\mathcal{M}, g)$ be a compact Riemannian manifold of dimension $N\geq 2$. We prove the existence of a family $(\Omega_\varepsilon)_{\varepsilon\in (0,\varepsilon_0)}$ of self-Cheeger sets in $(\mathcal{M}, g)$ . The domains $\Omega_\varepsilon\subset\mathcal{M}$ are perturbations of geodesic balls of radius $\varepsilon$ centered at $p \in \mathcal{M}$, and in particular, if $p_0$ is a non-degenerate critical point of the scalar curvature of $g$, then the family $( \partial\Omega_\varepsilon)_{\varepsilon\in (0,\varepsilon_0)}$ constitutes a smooth foliation of a neighborhood of $p_0$.