arXiv research

Synthetic splitting theorem for Lorentzian spaces with non-negative curvature.

The abstract proves a global splitting theorem for Poisson manifolds.

Paper proves new theorems about curvature in weighted manifolds.

Low regularity spacetimes split into simpler structures.

Splitting theorem for non-positively curved Lorentzian spaces.

New splitting theorem for weighted Finsler spacetimes without Berwald condition.

This paper generalizes Batchelor's theorem in $C^\infty$-superschemes.

Researchers prove a 30-year-old cosmological conjecture about spacetime.

We prove a Lorentzian splitting theorem with weakened curvature conditions.

Global rigidity theorem for certain lattice actions on manifolds.

The null splitting theorem (proved in math.DG/9909158) is discussed. As an application, a uniqueness theorem for Minkowski space and for de Sitter space associated with the occurrence of null lines (inextendible globally achronal null geodesics) is presented.

Geroch's theorem about the splitting of globally hyperbolic spacetimes is a central result in global Lorentzian Geometry. Nevertheless, this result was obtained at a topological level, and the possibility to obtain a metric (or, at least, smooth) version has been controversial since its publication in 1970. In fact, th…

Given a globally hyperbolic spacetime $M$, we show the existence of a {\em smooth spacelike} Cauchy hypersurface $S$ and, thus, a global diffeomorphism between $M$ and $\R \times S$.

The paper explores rigidity and splitting theorems for sub-static spaces with minimal hypersurfaces.

Introduces Alexandrov spaces with curvature below, covering various theorems.

This paper looks at the splitting problem for globally hyperbolic spacetimes with timelike Ricci curvature bounded below containing a (spacelike, acausal, future causally complete) hypersurface with mean curvature bounded from above. For such spacetimes we show a splitting theorem under the assumption of either the exi…

We prove a local splitting theorem for three-manifolds with mean convex boundary and scalar curvature bounded from below that contain certain locally area-minimizing free boundary surfaces. Our methods are based on those of Micallef and Moraru. We use this local result to establish a global rigidity theorem for area-mi…

In this paper we provide some local and global splitting results on complete Riemannian manifolds with nonnegative Ricci curvature. We achieve the splitting through the analysis of some pointwise inequalities of Modica type which hold true for every bounded solution to a semilinear Poisson equation. More precisely, we …

Study confirms Chern's conjecture on compact Hessian manifolds and classifies their topologies.

Inspired by the results in a recent paper by G. Galloway and C. Vega (see arXiv:1712.00785), we investigate a number of geometric consequences of the existence of a timelike conformal Killing vector field on a globally hyperbolic spacetime with compact Cauchy hypersurfaces, especially in connection with the so-called B…

We will show the following three theorems on the diffeomorphism and homeomorphism groups of a $K3$ surface. The first theorem is that the natural map $π_{0}(Diff(K3)) \to Aut(H^{2}(K3;\mathbb{Z}))$ has a section over its image. The second is that, there exists a subgroup $G$ of $π_{0}(Diff(K3))$ of order two over which…

New proof of Lorentzian splitting theorems using elliptic operators.

We analyze Lorentzian spacetimes subject to curvature-dimension bounds using the Bakry-Émery-Ricci tensor. We extend the Hawking-Penrose type singularity theorem and the Lorentzian timelike splitting theorem to synthetic dimensions $N\le 1$, including all negative synthetic dimensions. The rigidity of the timelike spli…

Study numerically flat foliations on Kähler manifolds, proving new splitting theorems.

In this paper, we study global properties of continuous plurisubharmonic functions on complete noncompact Kähler manifolds with nonnegative bisectional curvature and their applications to the structure of such manifolds. We prove that continuous plurisubharmonic functions with reasonable growth rate on such manifolds c…

Study splitting submanifolds in specific homogeneous spaces.

Paper establishes a new formula for Atiyah-Patodi-Singer index using eta invariants.

We investigate a generalization of the so-called metric splitting of globally hyperbolic space-times to non-smooth Lorentzian manifolds and show the existence of this metric splitting for a class of wave-type space-times. Our approach is based on smooth approximations of non-smooth space-times by families (or sequences…

Study Loday algebroids, prove splitting theorem, and linearize problems.

Study stability of surfaces in spacetimes, proving new estimates and theorems.

New theorem splits weighted Lorentz-Finsler manifolds into simpler parts.

Sharp spectral theorem splits certain non-compact manifolds.

We prove a splitting theorem for Riemannian n-manifolds with scalar curvature bounded below by a negative constant and containing certain area-minimising hypersurfaces (Theorem 3). Thus we generalise [25,Theorem 3] by Nunes. This splitting result follows from an area comparison theorem for hypersurfaces with non-positi…

Under an infinitesimal version of the Bishop-Gromov relative volume comparison condition for a measure on an Alexandrov space, we prove a topological splitting theorem of Cheeger-Gromoll type. As a corollary, we prove an isometric splitting theorem for Riemannian manifolds with singularities of nonnegative (Bakry-Emery…

New proof of splitting theorem and finite ends of minimal hypersurfaces in nonnegative curvature manifolds.

A maximum principle for C^0 null hypersurfaces is obtained and used to derive a splitting theorem for spacetimes which contain null lines. As a consequence of this null splitting theorem, it is proved that an asymptotically simple vacuum (Ricci flat) spacetime which contains a null line is isometric to Minkowski space.

The aim of this paper is to prove a normal form Theorem for Dirac-Jacobi bundles using the recent techniques from Bursztyn, Lima and Meinrenken. As the most important consequence, we can prove the splitting theorems of Jacobi pairs which was proposed by Dazord, Lichnerowicz and Marle. As an application we provide a alt…

Short proof of Strong Haken Theorem for 3-manifolds.

The paper classifies capillary graphs on manifolds with Ricci lower bounds.

This article concerns cotangent-lifted Lie group actions; our goal is to find local and ``semi-global'' normal forms for these and associated structures. Our main result is a constructive cotangent bundle slice theorem that extends the Hamiltonian slice theorem of Marle, Guillemin and Sternberg. The result applies to a…

We formulate extensions of Wilking's Jacobi field splitting theorem to uniformly positive sectional curvature and also to positive and nonnegative intermediate Ricci curvatures.

A theorem for debiasing machine learning with finite sample guarantees.

The paper proves a theorem about splitting manifolds with specific curvature properties.

Study on Kähler manifolds with non-negative mixed curvature, proving splitting and structure theorems.

In this paper, I prove a splitting theorem for equifocal submanifolds with non-flat section in a simply connected symmetric space of compact type. Also, by using the splitting theorem, I prove that the sections of equifocal submanifolds with non-flat section in an irreducible simply connected symmetric space of compact…

Abstract: Proves relative versions of group splitting results.

A Heegaard splitting of an open 3-manifold is the partition of the manifold into two non-compact handlebodies which intersect on their common boundary. This paper proves several non-compact analogues of theorems about compact Heegaard splittings. The main theorem is: if N is a compact, connected, orientable 3-manifold …

We study locally conformally Berwald metrics on closed manifolds which are not globally conformally Berwald. We prove that the characterization of such metrics is equivalent to characterizing incomplete, simply-connected, Riemannian manifolds with reducible holonomy group whose quotient by a group of homotheties is clo…