arXiv research

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

New examples challenge Geroch conjecture stability.

Study proves a conjecture for certain spin manifolds.

Modified condition proves no positive scalar curvature for enlargeable manifolds.

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…

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…

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.…

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

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…

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 $\R \times S$.

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.

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

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…

New PDE systems generalize Hawking mass monotonicity.

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.

Paper sharpens inequality linking curvature and spectrum on manifolds.

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…

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…

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

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

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

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…

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

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

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

Maximal spacetimes have unique past/future sets.

Survey on scalar curvature stability and related questions.

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

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

Upper estimate of Dirac eigenvalue linked to hyperspherical radius.

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…

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

Scalar curvature rigidity for products of convex hypersurfaces

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

This paper builds on the theory of generalised functions begun in [1]. The Colombeau theory of generalised scalar fields on manifolds is extended to a nonlinear theory of generalised tensor fields which is diffeomorphism invariant and has the sheaf property. The generalised Lie derivative for generalised tensor fields …

A knot $K$ is definite if $|σ(K)| = 2g(K)$. We prove that the quotient of a definite periodic knot is definite by considering equivariant minimal genus Seifert surfaces.

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…

A new geometric definition of integration for differential forms.

This study proves new financial market theorems breaking standard risk definitions.

The paper defines cross-section continuity for angular momentum definitions and finds the CWY definition valid.

Local positive definite Z2^n-superfunctions can be extended.

This paper proposes a new differential privacy definition using Rao distance.

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

New definition resolves ambiguity in non-stationary bandit classification.

Study negative definite spin fillings of knot covers.