Coulomb Problem in a One-Dimensional Space with Constant Positive Curvature

We construct a complex transformation generalizing the known Hurwitz transformation in the Euclidean space for one-dimensional quantum mechanics. As in the case of the flat space, this transformation allows establishing the connection between the Coulomb problem and the oscillator problem with the Calogero–Sutherland potential added. We fully describe the motion in a Coulomb field in S ₁ and determine the energy spectrum and the wave functions with the correct normalizing constant.     

Finsler Manifolds with Positive Constant Flag Curvature

It is shown that a Finsler metric with positive constant flag curvature and vanishing mean tangent curvature must be Riemannian. As applications, we also discuss the case of Cheng's maximal diameter theorem and Green's maximal conjugate radius theorem in Finsler manifolds.     

常曲率空间中的正曲率子流形

本文研究了常曲率空间子流表余维数减少的问题，说明了在一定条件下，余维数可以减少到１。     

The Problem of a Self-Gravitating Scalar Field with Positive Cosmological Constant

We study the Einstein-scalar field system with positive cosmological constant and spherically symmetric characteristic initial data given on a truncated null cone. We prove well-posedness, global existence and exponential decay in (Bondi) time, for small data. From this, it follows that initial data close enough to de Sitter data evolves to a causally geodesically complete spacetime (with boundary), which approaches a region of de Sitter asymptotically at an exponential rate; this is a non-linear stability result for de Sitter within the class under consideration, as well as a realization of the cosmic no-hair conjecture.     

The Cheeger constant of a quantum graph

We review the theory of Cheeger constants for graphs and quantum graphs and their present and envisaged applications. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Cheeger Constant, the Heat Kernel, and the Green Kernel of an Infinite Graph

 This paper gives a new approach to estimate the Cheeger constant, the heat kernel, and the Green kernel of the combinatorial Laplacian for an infinite graph. Received March 12, 2001; in final form July 11, 2002     

常曲率空间中具正Ricci曲率的子流形

设S~(n+p)(1)是一单位球面,M~n是浸入S~(n+p)(1)的具有非零平行平均曲率向量的n维紧致子流形.证明了当n≥4,p≥2时,如果M~n的Ricci曲率不小于(n-2)(1+H~2),则M~n是全脐的或者M~n的Ricci曲率等于(n-2)(1+H~2),进而M~n的几何分类被完全给出.     

Neumann Cheeger Constants on Graphs

For any subgraph of a graph, the Laplacian with Neumann boundary condition was introduced by Chung and Yau (Commun Anal Geom 2(4):627–640, 1994). In this paper, motivated by the Riemannian case, we introduce the Cheeger constants for Neumann problems and prove corresponding Cheeger estimates for first nontrivial eigenvalues.     

Differential as a harmonic morphism with respect to Cheeger–Gromoll-type metrics

We investigate horizontal conformality of a differential of a map between Riemannian manifolds, where the tangent bundles are equipped with Cheeger–Gromoll-type metrics. As a corollary, we characterize the differential of a map as a harmonic morphism.     

Asymptotics of Solutions of the Einstein Equations with Positive Cosmological Constant

A positive cosmological constant simplifies the asymptotics of forever expanding cosmological solutions of the Einstein equations. In this paper a general mathematical analysis on the level of formal power series is carried out for vacuum spacetimes of any dimension and perfect fluid spacetimes with linear equation of state in spacetime dimension four. For equations of state stiffer than radiation evidence for development of large gradients, analogous to spikes in Gowdy spacetimes, is found. It is shown that any vacuum solution satisfying minimal asymptotic conditions has a full asymptotic expansion given by the formal series. In four spacetime dimensions, and for spatially homogeneous spacetimes of any dimension, these minimal conditions can be derived for appropriate initial data. Using Fuchsian methods the existence of vacuum spacetimes with the given formal asymptotics depending on the maximal number of free functions is shown without symmetry assumptions.Communicated by Sergiu Klainermansubmitted 08/12/03, accepted 30/04/04     

The Cheeger constant of a Jordan domain without necks

We show that the maximal Cheeger set of a Jordan domain \(\Omega \) without necks is the union of all balls of radius \(r = h(\Omega )^{-1}\) contained in \(\Omega \). Here, \(h(\Omega )\) denotes the Cheeger constant of \(\Omega \), that is, the infimum of the ratio of perimeter over area among subsets of \(\Omega \), and a Cheeger set is a set attaining the infimum. The radius r is shown to be the unique number such that the area of the inner parallel set \(\Omega ^r\) is equal to \(\pi r^2\). The proof of the main theorem requires the combination of several intermediate facts, some of which are of interest in their own right. Examples are given demonstrating the generality of the result as well as the sharpness of our assumptions. In particular, as an application of the main theorem, we illustrate how to effectively approximate the Cheeger constant of the Koch snowflake.     

The Cheeger constant of curved tubes

We compute the Cheeger constant of tubular neighbourhoods of complete curves in an arbitrary dimensional Euclidean space and raise a question about curved spherical shells.     

Sharp Constant in a Jackson-Type Inequality for Approximation by Positive Linear Operators

In what follows, C is the space of -periodic continuous real-valued functions with uniform norm, is the first continuity modulus of a function with step h, H _n is the set of trigonometric polynomials of order at most n, is the set of linear positive operators (i.e., of operators such that for every ), is the space of square-integrable functions on ,

Constant Mean Curvature Surfaces of any Positive Genus

The paper shows the existence of several new families of noncompactconstant mean curvature surfaces: (i) singly punctured surfacesof arbitrary genus g1, (ii) doubly punctured tori, and (iii)doubly periodic surfaces with Delaunay ends.     

Uniqueness of the Cheeger set of a convex body

We prove that if C⊂R^N$C\subset {\mathbb{R}}^N$ is an open bounded convex set, then there is only one Cheeger set inside C$C$ and it is convex. A Cheeger set of C$C$ is a set which minimizes the ratio perimeter over volume among all subsets of C$C$.     

Erratum to: Paracontact Tangent Bundles with Cheeger–Gromoll Metric

Mediterranean Journal of Mathematics -