arXiv research

New examples challenge Geroch conjecture stability.

Proves conditions for positive scalar curvature on certain manifolds with conical singularities.

Paper reviews and proves the uniqueness of multipole moments for stationary spacetimes.

Study proves a conjecture for certain spin manifolds.

The goal of this paper is to establish the existence of a foliation of the asymptotic region of an asymptotically flat manifold with nonzero mass by surfaces which are critical points of the Willmore functional subject to an area constraint. Equivalently these surfaces are critical points of the Geroch-Hawking mass. Th…

Proves positive mass theorem for non-spin weighted manifolds.

For asymptotically flat initial data of Einstein's equations satisfying an energy condition, we show that the Penrose inequality holds between the ADM mass and the area of an outermost apparent horizon, if the data are restricted suitably. We prove this by generalizing Geroch's proof of monotonicity of the Hawking mass…

New curvature concept shows certain manifolds can't have positive curvature.

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 $\R \times S$.

Develops a method to deform metrics on manifolds with non-compact boundaries.

To cure the lack of predictive power of general relativity Geroch proposed to complete the theory with an additional postulate that only "hole-free" spacetimes are permitted. I argue that this postulate is too strong -- it prohibits even the Minkowski space.

New proof of Positive Mass Theorem using Green's function and monotonicity formula.

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…

The paper analyzes harmonic maps to flat model spaces to prove geometric stability of the positive mass theorem.

On the Geroch-Kronheimer-Penrose future completion $IP(X)$ of a spacetime $X$, there are two frequently used topologies. We systematically examine $τ_+$, the stronger (metrizable) of them, which is the coarsest causally continuous topology, obtaining a variety of novel results, among them a complete characterization of…

New PDE systems generalize Hawking mass monotonicity.

Examples of almost-positively and quasi-positively curved spaces of the form M=H((G,h)xF) were discovered recently. Here, h is a left-invariant metric on a compact Lie group G, F is a compact Riemannian manifold on which the subgroup H of G acts isometrically on the left, and M is the orbit space of the diagonal left a…

In 1969, Choquet-Bruhat and Geroch established the existence of a unique maximal globally hyperbolic Cauchy development of given initial data for the Einstein equations. Their proof, however, has the unsatisfactory feature that it relies crucially on the axiom of choice in the form of Zorn's lemma. In this paper we pre…

Proves mass theorem up to dimension 19 using symmetrization and singularity techniques.

The paper explores rigidity theorems for spectral curvature bounds in 3-manifolds.

Paper sharpens inequality linking curvature and spectrum on manifolds.

Modified condition proves no positive scalar curvature for enlargeable manifolds.

Defines timelike ideal boundary for non-positively curved Lorentzian spaces.

Maximal spacetimes have unique past/future sets.

Study establishes time functions in Lorentzian spaces without requiring manifold structure.

This paper completes globally hyperbolic conformally flat spacetimes, proving they are topological manifolds.

In a 2013 paper, Gromov proves that if smooth Riemannian metrics $g_i$ converge to a smooth Riemannian metric $g$ uniformly, and $g_i$ have scalar curvature uniformly bounded below, then $g$ shares the same scalar curvature lower bound. In some places in the paper, the proofs are only sketched. In this paper we explain…

New bounds for low-regularity Riemannian metrics defined via distributional curvature.

Establishes existence of maximal globally hyperbolic development for Einstein equations.

Following Geroch, Traschen, Mars and Senovilla, we consider Lorentzian manifolds with distributional curvature tensor. Such manifolds represent spacetimes of general relativity that possibly contain gravitational waves, shock waves, and other singular patterns. We aim here at providing a comprehensive and geometric (i.…

Study Gromov hyperbolic domains in Minkowski space, proving equivalence to boundary properties.

Proves non-existence of metrics with positive curvature for certain connected sums.

Survey on scalar curvature stability and related questions.

Upper estimate of Dirac eigenvalue linked to hyperspherical radius.

Scalar curvature rigidity for products of convex hypersurfaces

The notion of maximal extension of a globally hyperbolic space-time arises from the notion of maximal solutions of the Cauchy problem associated to the Einstein's equations of general relativity. In 1969 Choquet-Bruhat and Geroch proved that if the Cauchy problem has a local solution, this solution has a unique maximal…

Minimal TIP and TIF found in compact spacetimes, impacting spacetime splitting.

The study explores spacetimes with changing spatial curvature, leading to topological transitions.

Extends nonlinear theory of distributional geometry.

We show that when a spacetime $\mathcal{M}(=M \cup \partial M)$ is globally hyperbolic with (possibly empty) smooth timelike boundary $\partial M$, a metrizable topology, the closed limit topology (CLT) introduced by F. Hausdorff himself in the 1950's in set theory, can be advantageously adopted on the Geroch-Kronheime…

We study Ricci flows of some classes of physically valuable solutions in Einstein and string gravity. The anholonomic frame method is applied for generic off-diagonal metric ansatz when the field/ evolution equations are transformed into exactly integrable systems of partial differential equations. The integral varieti…

Proves initial data on big bang singularities for Einstein-nonlinear scalar field equations lead to unique solutions.

The paper proves rigidity results for Serrin-type problems in manifolds.

Study rigidifies Einstein-type manifolds with boundary and constant curvature.

Improved understanding of translating solitons using new techniques.

In this paper, by employ the cutoff function and the maximum principle, some Hamilton-Souplet-Zhang type gradient estimates for porous medium type equation are deduced. As a special case, an Hamilton-Souplet-Zhang type gradient estimates of the heat equation is derived which is different from the result of Souplet-Zhan…

We study the transfer of adversarial robustness of deep neural networks between different perturbation types. While most work on adversarial examples has focused on $L_\infty$ and $L_2$-bounded perturbations, these do not capture all types of perturbations available to an adversary. The present work evaluates 32 attack…

Two results on end spaces of infinite type surfaces, answering questions about their topology and equivalence.