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If $\varphi$ is not minimal (the mean curvature $H$ is not identically zero), then for every $f \in C^\infty(\Sigma)$ such that $\int_\Sigma f H \, \mathrm{d} vol_{\varphi^\ast g} = 0$ there exists a proper …

first eigenvalue of a small geodesic ball in a riemannian manifold”, by Karp and Pinsky, from where I took the following: Here, $\Delta_{-2}$ denotes the usual Laplacian in $\mathbb{R}^n$, $R$ is the curvature …

}t} \right\vert_{t=0} \mathcal{A}([f_t], g_t) = - \int_M H_{f} g(X, \nu) \, \mathrm{d}A_{f^\ast g} + \int_M \operatorname{tr}_{f^\ast g}(f^\ast h) \, \mathrm{d}A_{f^\ast g},$$ where $H_f$ is the mean curvature …

The following result was taken from the book of Gilbarg-Trudinger: In particular, if the graph is minimal, then $u$ is smooth. Now comes my question: does the same conclusion hold for graphs over dom …

Let $(M,g)$ be a complete and oriented Riemannian 3-manifold without boundary. Given $\Sigma_1$ and $\Sigma_2$ closed surfaces embedded in $M$, where the former is minimal and the latter has CMC $H > …

Assume it has positive scalar curvature and $\partial M$ is mean convex (positive mean curvature). Question: Is it true that $DM$, the double of $M$, admits a metric of positive scalar curvature? …

Let $(M,g)$ be a compact Riemannian manifold with boundary and assume it has positive scalar curvature. Question. … Is it true that $DM$, the double of $M$, admits a metric of positive scalar curvature? …

Is it true that if $D$ is a Riemannian $2$-disk having constant Gaussian curvature equal to $1$ and whose boundary has constant geodesic curvature, then $D$ is isometric to some geodesic ball of the unit …

\inf_{\partial M} H^{\partial M} \leq 2 \pi,$$ where $R_M$ is the scalar curvature of $M$ and $H^{\partial M}$ is the mean curvature of $\partial M$. … $\partial \Sigma_0$ has constant geodesic curvature $\inf_{\partial M} H^{\partial M}$ in $\Sigma_0$, does equality hold? …

Li proved a complete topological classification of those compact, connected and orientable $3$-manifolds with boundary which support Riemannian metrics of positive scalar curvature and mean-convex boundary …

Suppose $M$ has nonnegative Ricci curvature and the boundary $\partial M$ is strictly mean convex with respect to the inward unit normal. … My question is: what are examples of compact Riemannian $3$-manifolds with nonnegative scalar curvature (but not nonnegative Ricci curvature) and mean convex boundary that don't admit closed embedded minimal …

Is it known whether the scalar curvature of $(M,\tilde{g})$ can be strictly positive? …

A theorem by Eschenburg states that if $M$ is $\varepsilon$-convex and $N$ has nonnegative sectional curvature, then $M$ is the boundary of a convex body in $N$; in particular, $M$ is diffeomorphic to …

Let $(\overline{M}^{n+1}, \langle \cdot, \cdot \rangle)$ be a riemannian manifold with riemannian connection $\overline{\nabla}$ and consider $M^n \subset \overline{M}$ an orientable hypersurface with …

Suppose $M$ is contained in the northern hemisphere $S_+^{n+1}$ and has nonzero principal curvatures everywhere, i.e., has nonzero gaussian curvature everywhere. …