Related papers: Two rigidity results for surfaces in Schwarzschild…
The aim of this paper is to prove two results concerning the rigidity of complete, immersed, orientable, stable minimal hypersurfaces: we show that they are hyperplane in $\mathbb{R}^4$, while they do not exist in positively curved closed…
Several rigidity results are proved for critical points of natural Riemannian functionals on the space of metrics on 3-manifolds. Two of these results are as follows. Let (N, g) be a complete Riemannian 3-manifold, satisfying one of the…
We present a class of spherically symmetric hypersurfaces in the Kruskal extension of the Schwarzschild space-time. The hypersurfaces have constant negative scalar curvature, so they are hyperboloidal in the regions of space-time which are…
We examine some common features of minimal surfaces, nonzero constant mean curvature surfaces and marginally outer trapped surfaces, concerning their stability and rigidity, and consider some applications to Riemannian geometry and general…
For asymptotically flat spacetimes, using the inverse mean curvature flow, we show that any compact $2$-surface, $S_0$, whose mean curvature and its derivative for outward direction are positive in spacelike hypersurface with non-negative…
Exploiting the special features of four-dimensional Riemannian geometry, we derive topological and rigidity results for hypersurfaces immersed in space forms of dimension 5. First, we provide a complete description of the Weyl tensor for…
We prove that a spacelike spherical symmetric constant mean curvature (SSCMC) surface and a general spacelike constant mean curvature (CMC) surface with certain boundary condition at the future null-infinity in Schwarzschild spacetime are…
We identify, in spacetimes satisfying the null convergence condition, a certain natural class of null hypersurfaces that admit null sections with constant surface gravity. Our work is meant to offer complementary results to previous work on…
Motivated by the rigidity case in the localized Riemannian Penrose inequality, we show that suitable singular metrics attaining the optimal value in the Riemannian Penrose inequality is necessarily smooth in properly specified coordinates.…
Given a time symmetric initial data set for the vacuum Einstein field equations which is conformally flat near infinity, it is shown that the solutions to the regular finite initial value problem at spatial infinity extend smoothly through…
We study the geometry of stable maximal hypersurfaces in a variety of spacetimes satisfying various physically relevant curvature assumptions, for instance the Timelike Convergence Condition (TCC). We characterize stability when the target…
In this paper, we deduce some rigidity results in warped product spaces under normal variations of CMC hypersurfaces. In particular, we prove the existence of one-parameter families locally rigid on the spatial fiber of Anti-de Sitter…
We first demonstrate that the area preserving mean curvature flow of hypersurfaces in space forms exists for all time and converges exponentially fast to a round sphere if the integral of the traceless second fundamental form is…
Motivated by a result of Treibergs, given a smooth function f(y) on the standard sphere S^2, and any positive constant H_0, we construct a spacelike surface with constant mean curvature H_0 in the Schwarzschild spacetime, which is the graph…
A general geometrical scheme is presented for the construction of novel classical gravity theories whose solutions obey two-sided bounds on the sectional curvatures along certain subvarieties of the Grassmannian of two-planes. The…
Nontrivial infinitesimal bendings for a class of two-dimensional surfaces are constructed. The surfaces considered here are orientable; compact; with boundary; have positive curvature everywhere except at finitely many planar points; and…
A compactness theorem is proved for a family of K\"{a}hler surfaces with constant scalar curvature and volume bounded from below, diameter bounded from above, Ricci curvature bounded and the signature bounded from below. Furthermore, a…
We study the backwards-in-time stability of the Schwarzschild singularity from a dynamical PDE point of view. More precisely, considering a spacelike hypersurface $\Sigma_0$ in the interior of the black hole region, tangent to the singular…
We present several rigidity results for Riemannian manifolds $(M^n,g)$ with scalar curvature $S \ge -n(n-1)$ (or $S\ge 0$), and having compact boundary $N$ satisfying a related mean curvature inequality. The proofs make use of results on…
We investigate integral conditions involving the mean curvature vector $\vec{H}$ or mixed higher-order mean curvatures, to determine when a codimension-two submanifold $\Sigma$ lies on a shear-free (umbilical) null hypersurface in a…