### 0-tight surfaces with boundary and the total curvature of curves in surfaces

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We study the notion of strong $r$-stability for the context of closed hypersurfaces ${\Sigma}^{n}$ ($n\ge 3$) with constant $(r+1)$-th mean curvature ${H}_{r+1}$ immersed into the Euclidean sphere ${\mathbb{S}}^{n+1}$, where $r\in \{1,...,n-2\}$. In this setting, under a suitable restriction on the $r$-th mean curvature ${H}_{r}$, we establish that there are no $r$-strongly stable closed hypersurfaces immersed in a certain region of ${\mathbb{S}}^{n+1}$, a region that is determined by a totally umbilical sphere of ${\mathbb{S}}^{n+1}$. We also provide a rigidity result for such hypersurfaces.

In this paper we study the $r$-stability of closed hypersurfaces with constant $r$-th mean curvature in Riemannian manifolds of constant sectional curvature. In this setting, we obtain a characterization of the $r$-stable ones through of the analysis of the first eigenvalue of an operator naturally attached to the $r$-th mean curvature.