Related papers: A note on the area and coarea formulas for general…
We present symbolic and numerical methods for computing Poisson brackets on the spaces of measures with positive densities of the plane, the 2-torus, and the 2-sphere. We apply our methods to compute symplectic areas of finite regions for…
In this work, we consider the area non-increasing map between manifolds with positive curvature. By exploring the strong maximum principle along the graphical mean curvature flow, we show that an area non-increasing map between certain…
Lipid bilayer membranes are commonly modeled as area-preserving fluid surfaces that resist bending. There appear to be two schools of thought in the literature concerning the actual area constraint. In some works the total or global area…
This paper is devoted to the proof of a well-posedness result for the gravity water waves equations, in arbitrary dimension and in fluid domains with general bottoms, when the initial velocity field is not necessarily Lipschitz. Moreover,…
In this manuscript, we put forth a general scheme for defining initial value problems from Einstein's equations of General Relativity constrained by homogeneous and isotropic expansion. The cosmological models arising as solutions are…
The intention of this article is to give a flavour of some global problems in General Relativity. We cover a variety of topics, some of them related to the fundamental concept of 'Cauchy hypersurfaces': (1) structure of globally hyperbolic…
We establish two types of characterizations for high order anisotropic Sobolev spaces. In particular, we prove high order anisotropic versions of Bourgain-Brezis- Mironescu's formula and Nguyen's formula.
We find universal spaces for Alexandroff and finite spaces and explore some of its topological properties as well as their description as inverse limits of finite spaces and Alexandroff extensions. They can be used as a natural environment…
We prove a new inequality relating volume to length of closed geodesics on area minimizers for generic metrics on the complex projective plane. We exploit recent regularity results for area minimizers by Moore and White, and the…
We derive basic differential geometric formulae for surfaces in hyperbolic space represented as envelopes of horospheres. The dual notion of parallel hypersurfaces is also studied. The representation is applied to prove existence and…
We extend the theory of Patterson-Sullivan measure to any regular covering of a compact manifold using the Busemann compactification and derive an integral formula for the volume entropy. As applications we prove some rigidity theorems for…
We propose a reconstruction of general bulk surfaces in any dimension in terms of the differential entropy in the boundary field theory. In particular, we extend the proof of Headrick et al. to calculate the area of a general class of…
We prove new concentration estimates for random variables that are functionals of a Poisson measure defined on a general measure space. Our results are specifically adapted to geometric applications, and are based on a pervasive use of a…
In this article, we will use the harmonic mean curvature flow to prove a new class of Alexandrov-Fenchel type inequalities for strictly convex hypersurfaces in hyperbolic space in terms of total curvature, which is the integral of Gaussian…
We develop a theory for the representation of opaque solids as volumes. Starting from a stochastic representation of opaque solids as random indicator functions, we prove the conditions under which such solids can be modeled using…
The $p$-modulus of curves, test plans, upper gradients, charts, differentials, approximations in energy and density of directions are all concepts associated to the theory of Sobolev functions in metric measure spaces. The purpose of this…
We prove various inequalities measuring how far from an isometry a local map from a manifold of high curvature to a manifold of low curvature must be. We consider the cases of volume-preserving, conformal and quasi-conformal maps. The…
We develop an Aleksandrov reflection framework for a large class of expanding curvature flows in hyperbolic space, with inverse mean curvature flow serving as a model case. The method applies to the level-set formulation of the flow. As a…
Weighted cone-volume functionals are introduced for the convex polytopes in $\mathbb{R}^n$. For these functionals, geometric inequalities are proved and the equality conditions are characterized. A variety of corollaries are derived,…
We investigate two types of characterizations for anisotropic Sobolev and BV spaces. In particular, we establish anisotropic versions of the Bourgain-Brezis-Mironescu formula, including the magnetic case both for Sobolev and BV functions.