Existence of Self-Cheeger sets on Riemannian manifolds

Abstract

Let $(\mathcal{M},g)$ be a compact Riemannian manifold of dimension $N\ge 2$. We prove the existence of a family $({\Omega }_{\epsilon }{)}_{\epsilon \in (0,{\epsilon }_0)}$ of self-Cheeger sets in $(\mathcal{M},g)$ . The domains ${\Omega }_{\epsilon }\subset \mathcal{M}$ are perturbations of geodesic balls of radius $\epsilon$ 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 $(\partial {\Omega }_{\epsilon }{)}_{\epsilon \in (0,{\epsilon }_0)}$ constitutes a smooth foliation of a neighborhood of $p_0$.

Revised argument on section 2, Results unchanged. Published in Archiv der Mathematik

References in corpus (2)

• Continuity of the isoperimetric profile of a complete Riemannian manifold under sectional curvature conditions
• The Cheeger Constant, Isoperimetric Problems, and Hyperbolic Surfaces