Related papers: Rigidity results for complete spacelike submanifol…

Ratner's theorem implies topological rigidity of immersed totally geodesic subspaces of noncompact type in finite-volume locally symmetric spaces. In higher rank and infinite volume, however, counter-examples to this rigidity have remained…

We consider closed hypersurfaces smoothly immersed in hyperbolic manifolds up to homotopy and commensurability. We prove that if a closed hyperbolic manifold $M$ contains a sequence of asymptotically geodesic hypersurfaces, then $\pi_1(M)$…

In the study of immersed surfaces of constant positive extrinsic curvature in space-forms, it is natural to substitute completeness for a weaker property, which we here call quasicompleteness. We determine the global geometry of such…

For a given space-time and for an arbitrary time-like geodesic, we analyze the conditions for the construction of Fermi coordinates so that they are also rigid covariant. We then apply these conditions to linear plane gravitational waves.

We give a full description of totally geodesic submanifolds in the tangent bundle of a Riemannian 2-manifold of constant curvature and present a new class of a cylinder-type totally geodesic submanifolds in the general case.

We study complete spacelike hypersurfaces immersed in an open region of the de Sitter space $\mathbb{S}^{n+1}_{1}$ which is known as the steady state space $\mathcal{H}^{n+1}$. In this setting, under suitable constraints on the behavior of…

We investigate suitable, physically motivated conditions on spacetimes containing certain submanifolds - the so-called {weakly trapped submanifolds} - that ensure, in a set of neighboring metrics with respect to a convenient topology, that…

A general integral inequality is established for compact spacelike submanifolds of codimension two in the Lorentz-Minkowski spacetime under the assumption that the mean curvature vector field is parallel. This inequality is then used to…

This paper investigates the global properties of a class of spherically symmetric spacetimes. The class contains the maximal development of asymptotically flat spherically symmetric initial data for a wide variety of coupled Einstein-matter…

We present a quantitative isolation property of the lifts of properly immersed geodesic planes in the frame bundle of a geometrically finite hyperbolic $3$-manifold. Our estimates are polynomials in the tight areas and…

Based on properties of n-subharmonic functions we show that a complete, noncompact, properly embedded hypersurface with nonnegative Ricci curvature in hyperbolic space has an asymptotic boundary at infinity of at most two points. Moreover,…

In this paper, we perform a detailed investigation on the various geometrical properties of trapped surfaces and the boundaries of trapped region in general relativity. This treatment extends earlier work on LRS II spacetimes to a general 4…

For a Riemannian manifold $M$, possibly with boundary, we consider the Riemannian product $M\times\mathbb{R}^k$ with a smooth positive function that weights the Riemannian measures. In this work we characterize parabolic hypersurfaces with…

We prove rigidity results involving the Hawking mass for surfaces immersed in a $3$-dimensional, complete Riemannian manifold $(M,g)$ with non-negative scalar curvature (resp. with scalar curvature bounded below by $-6$). Roughly, the main…

For compact Riemannian manifolds with convex boundary, B.White proved the following alternative: Either there is an isoperimetric inequality for minimal hypersurfaces or there exists a closed minimal hypersurface, possibly with a small…

This note proves the geodesic completeness of any compact manifold endowed with a linear connection such that the closure of its holonomy group is compact.

We investigate the topology and geometry of compact submanifolds in space forms of nonnegative curvature that satisfy a lower bound on the sectional curvature, depending only on the length of the mean curvature vector of the immersion. We…

Let $\Gamma$ be a lattice in $\mathrm{SO}_0(n, 1)$. We prove that if the associated locally symmetric space contains infinitely many maximal totally geodesic subspaces of dimension at least $2$, then $\Gamma$ is arithmetic. This answers a…

As discussed in the paper, in a matter-filled spacetime, perhaps with positive cosmological constant, a stable marginally outer trapped 2-sphere must satisfy a certain area inquality. Namely, its area must be bounded above by $4\pi/c$,…

We investigate the appearance of closed timelike curves in quotients of plane waves along spacelike isometries. First we formulate a necessary and sufficient condition for a quotient of a general spacetime to preserve stable causality. We…