相关论文: Some geometric properties of generalized $\varphi$…
Given a closed Riemannian manifold of dimenion less than eight, we prove a compactness result for the space of closed, embedded minimal hypersurfaces satisfying a volume bound and a uniform lower bound on the first eigenvalue of the…
In this contribution, we study spacetimes of cosmological interest, without making any symmetry assumptions. We prove a rigid Hawking singularity theorem for positive cosmological constant, which sharpens known results. In particular, it…
In this paper we give pinching theorems for the first nonzero eigenvalue of the Laplacian on the compact hypersurfaces of ambient spaces with bounded sectional curvature. As application we deduce rigidity results for stable constant mean…
We prove the cosmic no-hair conjecture for all orthogonal Bianchi cosmologies with matter in the $R+\beta R^2$ theory using the conformally equivalent Einstein field equations, with the scalar field having the full self-interacting…
We study constant mean curvature spacelike hypersurfaces in generalized Robertson-Walker spacetimes which are spatially parabolic covered (i.e. its fiber F is a (non- compact) complete Riemannian manifold whose universal covering is…
Naturally reductive spaces, in general, can be seen as an adequate generalization of Riemannian symmetric spaces. Nevertheless, there are some that are closer to symmetric spaces than others. On the one hand, there is the series of Hopf…
We study the asymptotic stability of de Sitter spacetime with respect to non-linear perturbations, by considering second order perturbations of a flat Robertson-Walker universe with dust and a positive cosmological constant. Using the…
We establish the essentially optimal form of Donaldson's geodesic stability conjecture regarding existence of constant scalar curvature K\"ahler metrics. We carry this out by exploring in detail the metric geometry of Mabuchi geodesic rays,…
Curvature is a fundamental geometric characteristic of smooth spaces. In recent years different notions of curvature have been developed for combinatorial discrete objects such as graphs. However, the connections between such discrete…
We study the uniqueness of horospheres and equidistant spheres in hyperbolic space under different conditions. First we generalize the Bernstein theorem by Do Carmo and Lawson to the embedded hypersurfaces with constant higher order mean…
Static vacuum near horizon geometries are solutions $(M,g,X)$ of a certain quasi-Einstein equation on a closed manifold $M$, where $g$ is a Riemannian metric and $X$ is a closed 1-form. It is known that when the cosmological constant…
Searching for the dynamical foundations of the Havrda-Charv\'{a}t/Dar\'{o}czy/Cressie-Read/Tsallis non-additive entropy, we come across a covariant quantity called, alternatively, a generalized Ricci curvature, an $N$-Ricci curvature or a…
This is the first of two papers where we study the asymptotics of the generalized Nariai solutions and its relation to the cosmic no-hair conjecture. According to the cosmic no-hair conjecture, generic expanding solutions of Einstein's…
In this paper we study the spacelike-characteristic Cauchy problem for the Einstein vacuum equations. We prove that given initial data on a maximal compact spacelike hypersurface $\Sigma \simeq \overline{B(0,1)} \subset \mathbb{R}^3$ and…
Several proposals for Quantum Gravity involve length and area operators with discrete eigenvalues. I show that the analyses of some simple procedures for the measurement of areas and lengths suggest that this discreteness characterizing the…
A new functional calculus, developed recently for a fully non-perturbative treatment of quantum gravity, is used to begin a systematic construction of a quantum theory of geometry. Regulated operators corresponding to areas of 2-surfaces…
We discuss the no-hair conjecture in the presence of a cosmological constant. For the firststep the real scalar field is considered as the matter field and the spacetime is assumed to be static spherically symmetric. If the scalar field is…
An exact solution to the vacuum Einstein equations is presented, whose structure is based on the Hopf fibration. The solution employs a geodesic null vector field that defines a twisting congruence and appears in the metric in Kerr-Schild…
In the first part of this paper we consider expanding vacuum cosmological spacetimes with a free $T^N$-action. Among them, we give evidence that Gowdy spacetimes have AVTD (asymptotically velocity term dominated) behavior for their initial…
We study the quantization of geometry in the presence of a cosmological constant, using a discretiza- tion with constant-curvature simplices. Phase space turns out to be compact and the Hilbert space finite dimensional for each link. Not…