arXiv subjects

Abraão Mendes

Publications and source records attributed to Abraão Mendes.

22 records · Page 2Linked to original sources

arXiv2018

Rigidity of volume-minimizing hypersurfaces in Riemannian 5-manifolds

In this paper we generalize the main result of [4] for manifolds that are not necessarily Einstein. In fact, we obtain an upper bound for the volume of a locally volume-minimizing closed hypersurface $Σ$ of a Riemannian 5-manifold $M$ with scalar curvature bounded from below by a positive constant in terms of the total traceless Ricci curvature of $Σ$. Furthermore, if $Σ$ saturates the respective upper bound and $M$ has nonnegative Ricci curvature, then $Σ$ is isometric to $\mathbb{S}^4$ up to scaling and $M$ splits in a neighborhood of $Σ$. Also, we obtain a rigidity result for the Riemannian cover of $M$ when $Σ$ minimizes the volume in its homotopy class and saturates the upper bound.

math.DG↗

arXiv2017

Rigidity of Free Boundary Surfaces in Compact 3-Manifolds with Strictly Convex Boundary

In this paper we obtain an analogue of Toponogov theorem in dimension 3 for compact manifolds $M^3$ with nonnegative Ricci curvature and strictly convex boundary $\partial M$. Here we obtain a sharp upper bound for the length $L(\partialΣ)$ of the boundary $\partialΣ$ of a free boundary minimal surface $Σ^2$ in $M^3$ in terms of the genus of $Σ$ and the number of connected components of $\partialΣ$, assuming $Σ$ has index one. After, under a natural hypothesis on the geometry of $M$ along $\partial M$, we prove that if $L(\partialΣ)$ saturates the respective upper bound, then $M^3$ is isometric to the Euclidean 3-ball and $Σ^2$ is isometric to the Euclidean disk. In particular, we get a sharp upper bound for the area of $Σ$, when $M^3$ is a strictly convex body in $\mathbb R^3$, which is saturated only on the Euclidean 3-balls (by the Euclidean disks). We also consider similar results for stationary stable surfaces.

math.DG↗

arXiv2016

Rigidity of Marginally Outer Trapped (Hyper)Surfaces with Negative $σ$-Constant

In this paper we generalize the main result of [13] in two different situations: in the first case for MOTSs of genus greater than one and, in the second case, for MOTSs of high dimension with negative $σ$-constant. In both cases we obtain a splitting result for the ambient manifold when it contains a stable closed MOTS which saturates a lower bound for the area (in dimension 2) or for the volume (in dimension $\ge3$). These results are extensions of [21, Theorem 3] and [20, Theorem 3] to general (non-time-symmetric) initial data sets.

math.DG↗

arXiv2015

Rigidity of marginally outer trapped 2-spheres

In a recent work, Galloway [9] proved a local foliation theorem by MOTSs for a 3-dimensional initial data set $(M,g,K)$ with mean curvature $τ\le0$ in a 4-dimensional spacetime $(\overline M,\overline g)$ when (under suitable assumptions) $M$ has a stable spherical MOTS $Σ$ which achieves an upper bound for the area. He proved that each leaf is a round 2-sphere with the same constant Gaussian curvature. Here, we improve his result by dropping the hypothesis on the mean curvature of $M$ and showing that each leaf is a minimal surface. We show that in an outer neighborhood $U$ of $Σ$, the metric $g$ splits as the product $([0,\varepsilon)\timesΣ,dt^2+g_0)$, where $(Σ,g_0)$ is a round 2-sphere. In fact, we prove that the outer neighborhood $U$ can be isometrically embedded into the 4-dimensional Nariai spacetime $(\overline N,\overline h)$ as a spacelike hypersurface so that $g|_U$ is the induced metric from $\overline N$ and $K|_U$ is the second fundamental form of $U$ in $\overline N$. This completely classifies the local geometry of such initial data sets.

math.DG↗