Related papers: Equidistant sets on Alexandrov surfaces

We provide a quick proof of the following known result: the Sobolev space associated with the Euclidean space, endowed with the Euclidean distance and an arbitrary Radon measure, is Hilbert. Our new approach relies upon the properties of…

We study compact stable embedded minimal surfaces whose boundary is given by two collections of closed smooth Jordan curves in close planes of Euclidean 3-space. Our main result is a classification of these minimal surfaces, under certain…

In this paper we generalize a result in [1], showing that an arbitrary Riemannian symmetric space can be realized as a closed submanifold of a covering group of the Lie group defining the symmetric space. Some properties of the subgroups of…

We study surfaces in Euclidean space constructed by the sum of two curves or that are graphs of the product of two functions. We consider the problem to determine all these surfaces with constant Gauss curvature. We extend the results to…

Compact flat surfaces of homogeneous Riemannian 3-manifolds with isometry group of dimension 4 are classified. Non-existence results for compact constant Gauss curvature surfaces in these 3-manifolds are established.

We prove that knowing the length of geodesics joining points on the boundary of a two-dimensional, compact, simple Riemannian manifold with boundary, we can determine uniquely the Riemannian metric up to the natural obstruction.

This paper provides a systematic exposition of Lyapunov stability for compact sets in locally compact metric spaces. We explore foundational concepts, including neighborhoods of compact sets, invariant sets, and the properties of dynamical…

Petrovskii elliptic systems of linear differential equations given on a closed smooth manifold are investigated on the extended Sobolev scale. This scale consists of all Hilbert spaces that are interpolation spaces with respect to the…

In this paper we provide a framework for the study of isoperimetric problems in finitely generated group, through a combinatorial study of universal covers of compact simplicial complexes. We show that, when estimating filling functions,…

For collapsing sequences of Riemannian manifolds which satisfy a uniform lower Ricci curvature bound it is shown that there is a sequence of scales such that for a set of good base points of large measure the pointed rescaled manifolds…

Geodesically complete affine manifolds are quotients of the Euclidean space through a properly discontinuous action of a subgroup of affine Euclidean transformations. An equivalent definition is that the tangent bundle of such a manifold…

A classical theorem of A.D. Alexandrov says that a connected compact smooth hypersurface in Euclidean space with constant mean curvature must be a sphere. We give exposition to some results on symmetry properties of hypersurfaces with…

We prove a number of dualities between posets and (pseudo)bases of open sets in locally compact Hausdorff spaces. In particular, we show that (1) Relatively compact basic sublattices are finitely axiomatizable. (2) Relatively compact basic…

We give new estimates for the extrinsic radius of compact hypersurfaces of the Euclidean space and the open hemisphere in terms of high order mean curvatures. Then we prove pinching results corresponding to theses estimates. We show that…

The paper investigates the (non)existence of compact quotients, by a discrete subgroup, of the homogeneous almost-complex strongly-pseudoconvex manifolds disconvered and classified by Gaussier-Sukhov and K.-H. Lee.

For a smooth surface in $\mathbb{R}^3$ this article contains local study of certain affine equidistants, that is loci of points at a fixed ratio between points of contact of parallel tangent planes (but excluding ratios 0 and 1 where the…

We obtain a structure theorem for closed, cohomogeneity one Alexandrov spaces and we classify closed, cohomogeneity one Alexandrov spaces in dimensions 3 and 4. As a corollary, we obtain the classification of closed, $n$-dimensional,…

We consider a range of geometric stability problems for hypersurfaces of spaceforms. One of the key results is an estimate relating the distance to a geodesic sphere of an embedded hypersurface with integral norms of the traceless Hessian…

Recently, much interest was devoted to the Urysohn universal metric space U and its isometries; this paper is a contribution to this field of research. In particular, we study some properties of isometries of U, and prove the following…

We prove that for a compact subgroup $H$ of a locally compact Hausdorff group $G$, the following properties are mutually equivalent: (1) $G/H$ is a manifold, (2) $G/H$ is finite-dimensional and locally connected, (3) $G/H$ is locally…