In a rotationally symmetric space $\oM$ around an axis A (whose precise definition includes all real space forms), we consider a domain $G$ limited by two equidistant hypersurfaces orthogonal to A. Let $M \subset \oM$ be a revolution hypersurface generated by a graph over A, with boundary in $\partial G$ and orthogonal to it. We study the evolution $M_t$ of $M$ under the volume-preserving mean curvature flow requiring that the boundary of $M_t$ rests on $\partial G$ and keeps orthogonal to it. We prove that: a) the generating curve of $M_t$ remains a graph; b) the flow exists while $M_t$ does not touch the axis of rotation; c) under a suitable hypothesis relating the enclosed volume and the area of $M$, the flow is defined for every $t\in [0,\infty[$ and a sequence of hypersurfaces $M_{t_n}$ converges to a revolution hypersurface of constant mean curvature. Some key points are: i) the results are true even for ambient spaces with positive curvature, ii) the averaged mean curvature does not need to be positive and iii) for the proof it is necessary to carry out a detailed study of the boundary conditions.
In a round ball $B\subset\mathbb{R}^{n+1}$ endowed with an $O(n+1)$-invariant metric we consider a radial function that weights volume and area. We prove that a compact two-sided hypersurface in $B$ which is stable capillary in weighted sense and symmetric about some line containing the center of $B$ is homeomorphic to a closed $n$-dimensional disk. When combined with Hsiang symmetrization and other stability results this allows to deduce that the interior boundary of any isoperimetric region in $B$ for the Gaussian weight is a closed $n$-disk of revolution. For $n=2$ we also show that a compa…
In the first part of this paper, we consider a partially overdetermined mixed boundary value problem in space forms and generalize the main result in \cite{GX} into the case of general domains with partial umbilical boundary in space forms. Precisely, we prove that a partially overdetermined problem in a domain with partial umbilical boundary admits a solution if and only if the rest part of the boundary is also part of an umbilical hypersurface. In the second part of this paper, we prove a Heintze-Karcher-Ros type inequality for embedded hypersurfaces with free boundary lying on a horosphere …
In this work we present a new family of properly embedded free boundary minimal hypersurfaces of revolution with circular boundaries in the horizon of the $n$-dimensional Schwarzschild space, $n\geq3$. In particular, we answer a question proposed by O. Chodosh and D. Ketover \cite{CK} on the existence of non-totally geodesic minimal surfaces in the 3-dimensional Schwarzschild space.
We obtain upper and lower bounds for Steklov eigenvalues of submanifolds with prescribed boundary in Euclidean space. A very general upper bound is proved, which depends only on the geometry of the fixed boundary and on the measure of the interior. Sharp lower bounds are given for hypersurfaces of revolution with connected boundary: we prove that each eigenvalue is uniquely minimized by the ball. We also observe that each surface of revolution with connected boundary is isospectral to the disk.
Let $Ω$ be a compact Riemannian manifold with smooth boundary and let $u_t$ be the solution of the heat equation on $Ω$, having constant unit initial data $u_0=1$ and Dirichlet boundary conditions ($u_t=0$ on the boundary, at all times). If at every time $t$ the normal derivative of $u_t$ is a constant function on the boundary, we say that $Ω$ has the {\it constant flow property}. This gives rise to an overdetermined parabolic problem, and our aim is to classify the manifolds having this property. In fact, if the metric is analytic, we prove that $Ω$ has the constant flow property if and only …
