arXiv research

Proves homological inequality for cycles in Hadamard spaces of asymptotic rank 2.

Sharp inequalities for manifolds with nonnegative curvature.

Sharp Sobolev and Michael-Simon inequalities on curved manifolds.

We construct transformations which take asymptotically AdS hyperbolic initial data into asymptotically flat initial data, and which preserve relevant physical quantities. This is used to derive geometric inequalities in the asymptotically AdS hyperbolic setting from counterparts in the asymptotically flat realm, whenev…

Ancient Ricci flows with asymptotic solitons have uniform bounds and inequalities.

Proves Sobolev inequality on manifolds with specific curvature properties.

In this note, we consider the isoperimetric inequality on an asymptotically flat manifold with nonnegative scalar curvature, and improve it by using Hawking mass. We also obtain a rigidity result when equality holds for the classical isoperimetric inequality on an asymptotically flat manifold with nonnegative scalar cu…

Willmore-type inequalities for bounded domains in manifolds with curvature bounds.

Proves a sharp inequality for toroidal surfaces in Horowitz-Myers geon.

Sharp inequality found for hypersurfaces in curved spaces.

Proves Penrose inequality for cohomogeneity one initial data sets.

The paper establishes inequalities for $p$-capacitary functions in flat half-spaces.

Sharp inequalities for $J$ functional on Kahler metrics.

Log-Sobolev inequality proven for submanifolds in specific types of manifolds.

The paper studies heat kernel asymptotics and proves Morse inequalities.

In this article, we prove the Riemannian Penrose inequality for asymptotically flat manifolds with non-compact boundary whose asymptotic region is modelled on a half-space. Such spaces were initially considered by Almaraz, Barbosa and de Lima in 2014. In order to prove the inequality, we develop a new approximation sch…

Sharp inequality linking interior and boundary Yamabe invariants on specific manifolds.

Researchers prove a Penrose inequality for spacetime with specific conditions.

Paper uses ABP method to prove logarithmic Sobolev inequalities on curved spaces.

The Minkowski inequality is a classical inequality in differential geometry, giving a bound from below, on the total mean curvature of a convex surface in Euclidean space, in terms of its area. Recently there has been interest in proving versions of this inequality for manifolds other than R^n; for example, such an ine…

Paper simplifies concentration inequalities for easier probabilistic analysis.

We construct a solution to inverse mean curvature flow on an asymptotically hyperbolic 3-manifold which does not have the convergence properties needed in order to prove a Penrose--type inequality. This contrasts sharply with the asymptotically flat case. The main idea consists in combining inverse mean curvature flow …

New proof of Penrose inequality using potential theory.

The paper consists of two parts. In the first part, by using the Gauss-Bonnet curvature, which is a natural generalization of the scalar curvature, we introduce a higher order mass, the Gauss-Bonnet-Chern mass $m^{\H}_k$, for asymptotically hyperbolic manifolds and show that it is a geometric invariant. Moreover, we pr…

We establish versions of the Positive Mass and Penrose inequalities for a class of asymptotically hyperbolic hypersurfaces. In particular, under the usual dominant energy condition, we prove in all dimensions $n\geq 3$ an optimal Penrose inequality for certain graphs in hyperbolic space $\mathbb H^{n+1}$ whose boundary…

We consider several geometric inequalities in general relativity involving mass, area, charge, and angular momentum for asymptotically hyperboloidal initial data. We show how to reduce each one to the known maximal (or time symmetric) case in the asymptotically flat setting, whenever a geometrically motivated system of…

Proves Riemannian Penrose Inequality for specific manifolds.

Study shows mass-capacity inequality for specific geometric manifolds.

We establish the inequality for Henneaux-Teitelboim's total energy-momentum for asymptotically anti-de Sitter initial data sets which are asymptotic to arbitrary $t$-slice in anti-de Sitter spacetime. In particular, when $t=0$, it generalizes Chruściel-Maerten-Tod's inequality in the center of AdS mass coordinates. We …

Proves equality in Minkowski inequality for static, flat manifolds.

Proves existence of solutions to Einstein constraints with specific boundary conditions and verifies Penrose inequality.

The study proves inequalities on curved spaces without global curvature bounds.

This study proves energy bounds in specific AdS spacetimes.

We prove the equality case of the Penrose inequality in all dimensions for asymptotically flat hypersurfaces. It was recently proven by G. Lam that the Penrose inequality holds for asymptotically flat graphical hypersurfaces in Euclidean space with non-negative scalar curvature and with a minimal boundary. Our main the…

We give estimates on asymptotic dimensions of products of general hyperbolic spaces with following applications to the hyperbolic groups. We give examples of strict inequality in the product theorem for the asymptotic dimension in the class of the hyperbolic groups; and examples of strict inequality in the product theo…

In this work, we prove an optimal Penrose inequality for asymptotically locally hyperbolic manifolds which can be realized as graphs over Kottler space. Such inequality relies heavily on an optimal weighted Alexandrov-Fenchel inequality for the mean convex star shaped hypersurfaces in Kottler space.

Proves Penrose inequality in all dimensions for specific manifolds.

We use the inverse mean curvature flow to prove a sharp Alexandrov-Fenchel-type inequality for a class of hypersurfaces in certain locally hyperbolic manifolds. As an application we derive an optimal Penrose inequality for asymptotically locally hyperbolic graphs in any dimension $n\geq 3$. When the horizon has the top…

Motivated by the equation satisfied by the extremals of certain Hardy-Sobolev type inequalities, we show sharp $L^q$ regularity for finite energy solutions of p-laplace equations involving critical exponents and possible singularity on a sub-space of $\mathbb{R}^n$, which imply asymptotic behavior of the solutions at i…

This paper constructs charged Riemannian manifolds to test Penrose inequality.

New mass definition linked to ADM mass for general metrics.

Study on charged parallel spinors and mass-charge inequalities.

In the asymptotically locally hyperbolic setting it is possible to have metrics with scalar curvature at least -6 and negative mass when the genus of the conformal boundary at infinity is positive. Using inverse mean curvature flow, we prove a Penrose inequality for these negative mass metrics. The motivation comes fro…

In this paper, we establish sharp inequalities for four kinds of classical eigenvalues on a bounded domain of a Riemannian manifold. We also establish asymptotic formulas for the eigenvalues of the buckling and clamped plate problems. In addition, we give a negative answer to the Payne conjecture for the one-dimensiona…

The paper proves $L^p$-Sobolev inequalities for minimal submanifolds.

We consider an asymptotically flat Lorentzian manifold of dimension (1,3). An inequality is derived which bounds the Riemannian curvature tensor in terms of the ADM energy in the general case with second fundamental form. The inequality quantifies in which sense the Lorentzian manifold becomes flat in the limit when th…

The paper proves isoperimetric inequalities in manifolds with small negative Ricci curvature.

The paper connects eigenvalue problems for various operators and establishes inequalities and asymptotic formulas for heat traces.