Existence of Self-Cheeger Sets on Riemannian Manifolds

Abstract

Let $(\mathcal{M}, g)$ be a compact Riemannian manifold of dimension $N\geq 2$. We prove the existence of a family $(Ω_\varepsilon)_{\varepsilon\in (0,\varepsilon_0)}$ of self-Cheeger sets in $(\mathcal{M}, g)$ . The domains $Ω_\varepsilon\subset\mathcal{M}$ 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 $( \partialΩ_\varepsilon)_{\varepsilon\in (0,\varepsilon_0)}$ constitutes a smooth foliation of a neighborhood of $p_0$.