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 $p_0$.
Categories: math.DG