Related papers: A note on the area and coarea formulas for general…
We study the Crofton's formula in the Lorentzian AdS$_3$ and find that the area of a generic space-like two dimensional surface is given by the flux of space-like geodesics. The "complexity=volume" conjecture then implies a new holographic…
We prove that a generic area-preserving diffeomorphism of a compact surface with non-empty boundary has an equidistributed set of periodic orbits. This implies that such a diffeomorphism has a dense set of periodic points, although we also…
We provide a family of global weighted Sobolev inequalities and Hardy inequalities on PI spaces with possibly non-maximal volume growth. Our results apply notably to non-trivial Ahlfors regular spaces like Laakso spaces and Kleiner-Schioppa…
We prove a universal coefficients theorem for the overconvergent cohomology modules introduced by Ash and Stevens, and give several applications. In particular, we sketch a very simple construction of eigenvarieties using overconvergent…
We apply the resummed version of the L\"uscher formula to analyze finite volume corrections to the mass of the nucleon and of heavy mesons. We show that by applying the subthreshold expansion of the scattering amplitudes one can express the…
Based on a novel type of Sobolev-Poincar\'e inequality (for generalised weakly differentiable functions on varifolds), we establish a finite upper bound of the geodesic diameter of generalised compact connected surfaces-with-boundary of…
The general form of the anisotropy parameter of the expansion for Bianchi type-III metric is obtained in the presence of a single diagonal imperfect fluid with a dynamically anisotropic equation of state parameter and a dynamical energy…
This paper studies Newtonian Sobolev-Lorentz spaces. We prove that these spaces are Banach. We also study the global p,q-capacity and the p,q-modulus of families of rectifiable curves. Under some additional assumptions (that is, the space…
In this paper, we obtain a new Hsiung-Minkowski integral formula for anisotropic capillary hypersurfaces in the half-space, which includes the weighted Hsiung-Minkowski formula and classical anisotropic Minkowski identity for closed…
The Cauchy problem for nonlinear elastic wave equations with viscoelastic damping terms is investigated in $L^{p}$ framework. It is proved that the small global solutions constructed in $L^{2}$-Sobolev spaces in our preceding paper [12]…
We obtain a Chern-Osserman type equality of a complete properly immersed surface in Euclidean space, provided the L^2-norm of the second fundamental form is finite. Also, by using a monotonicity formula, we prove that if the L^2-norm of…
We first review the $L^2$ bilinear generalizations of the $L^4$ estimate of Strichartz for solutions of the homogeneous 3D wave equation, and give a short proof based solely on an estimate for the volume of intersection of two thickened…
We present an algorithm to generalize a plethora of well-known solutions to Einstein field equations describing spherically symmetric relativistic fluid spheres by relaxing the pressure isotropy condition on the system. By suitably fixing…
We show that Bowen's equation, which characterises the Hausdorff dimension of certain sets in terms of the topological pressure of an expanding conformal map, applies in greater generality than has been heretofore established. In…
It is well known that finite-dimensional polyhedral convex sets can be generated by finitely many points and finitely many directions. Representation formulas in this spirit are obtained for convex polyhedra and generalized convex polyhedra…
Given an affine variety X and a finite dimensional vector space of regular functions L on X, we associate a convex body to (X, L) such that its volume is responsible for the number of solutions of a generic system of functions from L. This…
We employ methods of geometry and generalized convergence to construct a geometric measure that serves as an alternative to the integer-dimension Hausdorff measure. This construction prioritizes integration, yields the Area Formula as a…
In this paper we give characterizations of mappings generate embeddings of Sobolev spaces in the terms of ring capacity inequalities. In addition we prove that such mappings are Lipschitz mappings in the sub-hyperbolic type capacitory…
In this paper, the pinching problems of complete $\lambda$-hypersurfaces in a Euclidean space $\mathbb R^{n+1}$ are studied. By making use of the Sobolev inequality, we prove a global pinching theorem of complete $\lambda$-hypersurfaces in…
Consider a 2-plane $P \subset \mathbb{C}^n$ and let $D$ be a bounded region in $P$ with a piecewise-smooth boundary. Let $I(D)$ be the infimum of areas of all piecewise-smooth isotropic surfaces in $\mathbb{C}^n$ with the same boundary as…