Related papers: The isoperimetric problem for regular and crystall…

We consider a surface $M$ immersed in $\mathbb{R}^3$ with induced metric $g=\psi\delta_2$ where $\delta_2$ is the two dimensional Euclidean metric. We then construct a system of partial differential equations that constrain $M$ to lift to a…

In this article we establish $C^{3,\alpha}$-regularity of the reduced boundary of stationary points of a nonlocal isoperimetric problem in a domain $\Omega \subset \mathbb{R}^n$. In particular, stationary points satisfy the corresponding…

Suppose $ F $ is an integrand associated with a uniformly convex $ \mathscr{C}^{3} $-norm, and $ V $ is a $ n $-dimensional varifold in an open subset of $ \mathbf{R}^{n+1} $ such that $ \mathscr{H}^n \llcorner \operatorname{spt} \| V \| $…

Let $n\ge2$ and let $\Phi\colon\mathbb{R}^n\to[0,\infty)$ be a positively $1$-homogeneous and convex function. Given two convex bodies $A\subset B$ in $\mathbb{R}^n$, the monotonicity of anisotropic $\Phi$-perimeters holds, i.e.…

The Sobolev regularity of invariant measures for diffusion processes is proved on non-smooth metric measure spaces with synthetic lower Ricci curvature bounds. As an application, the symmetrizability of semigroups is characterized, and the…

We consider the inverse mean curvature flow by parallel hypersurfaces in space forms. We show that such a flow exists if and only if the initial hypersurface is isoparametric. The flow is characterized by an algebraic equation satisfied by…

We give a unified method for the general equivalence problem of extrinsic geometry, on the basis of our formulation of a general extrinsic geometry as that of an osculating map $\varphi\colon (M,\mathfrak f) \to L/L^0 \subset…

We classify all homothetical surfaces with constant mean curvature $H$ in the hyperbolic space $\mathbb{H}^3$. Using the upper half-space model with standard coordinates $(x,y,z)$, these surfaces are defined by the relation $z =…

We interpret the Hilbert entropy of a convex projective structure on a closed higher-genus surface as the Hausdorff dimension of the non-differentiability points of the limit set in the full flag space $\mathcal F(\mathbb R^3)$.…

In this paper, we investigate the minimization of a functional in which the usual perimeter is competing with a nonlocal singular term comparable (but not necessarily equal to) a fractional perimeter. The motivation for this problem is a…

The nonlinear wave equation $u_{tt}-c(u)(c(u)u_x)_x=0$ determines a flow of conservative solutions taking values in the space $H^1(\mathbb{R})$. However, this flow is not continuous w.r.t. the natural $H^1$ distance. Aim of this paper is to…

In the sub-Riemannian Heisenberg group equipped with its Carnot-Caratheodory metric and with a Haar measure, we consider isodiametric sets, i.e. sets maximizing the measure among all sets with a given diameter. In particular, given an…

It is well known that isoperimetric inequalities imply in a very general measure-metric-space setting appropriate concentration inequalities. The former bound the boundary measure of sets as a function of their measure, whereas the latter…

This paper deals with a variation of the classical isoperimetric problem in dimension $N\ge 2$ for a two-phase piecewise constant density whose discontinuity interface is a given hyperplane. We introduce a weighted perimeter functional with…

In general relativity, spatial light rays of static spherically symmetric spacetimes are geodesics of surfaces in Riemannian optical geometry. In this paper, we apply results on the isoperimetric problem to show that length-minimizing…

In this article, we consider compact Riemannian 3-manifolds with boundary. We prove that if the $L^2$-norm of the curvature is small and if the $H^{1/2}$-norm of the difference of the fundamental forms of the boundary is small, then the…

In this paper, the isoperimetric problem in the 2-dimensional Finsler space form $(F_B,B^2(1))$ with k = 0 by using the Busemann-Hausdorff area is investigated. We prove that the circle centered the origin achieves the local maximum area of…

Let $D$ be a closed disk centered at the origin in the horizontal hyperplane $\{t=0\}$ of the sub-Riemannian Heisenberg group $\hh^n$, and $C$ the vertical cylinder over $D$. We prove that any finite perimeter set $E$ such that $D\subset…

We use theory and numerical computation to determine the shape of an axisymmetric fluid membrane with a resistance to bending and constant area. The membrane connects two rings in the classic geometry that produces a catenoidal shape in a…

We explore a natural generalization of systolic geometry to Finsler metrics and optical hypersurfaces with special emphasis on its relation to the Mahler conjecture and the geometry of numbers. In particular, we show that if an optical…