We study the isoperimetric problem for Euclidean space endowed with a continuous density. In dimension one, we characterize isoperimetric regions for a unimodal density. In higher dimensions, we prove existence results and we derive stability conditions, which lead to the conjecture that for a radial log-convex density, balls about the origin are isoperimetric regions. Finally, we prove this conjecture and the uniqueness of minimizers for the density $\exp (|x|^2)$ by using symmetrization techniques.
For any closed Riemannian manifold $X$ we prove that large isoperimetric regions in $X\times{\mathbb R}^n$ are of the form $X\times$(Euclidean ball). We prove that if $X$ has non-negative Ricci curvature then the only soap bubbles enclosing a large volume are the products $X\times$(Euclidean sphere). We give an example of a surface $X$, with Gaussian curvature negative somewhere, such that the product $X\times{\mathbb R}$ contains stable soap bubbles of arbitrarily large enclosed volume which do not even project surjectively onto the $X$ factor.
We prove an sharp anisotropic isoperimetric inequality for a domain outside an Euclidean ball in $\mathbb{R}^n$ for $n\geq 2$. The proof applies the ABP method to a Neumann boundary value problem.
In this paper, we prove some isoperimetric bounds for lower order eigenvalues of the Wentzell-Laplace operator on bounded domains of a Euclidean space or a Hadamard manifold, of the Laplacian on closed hypersurfaces of a Euclidean space or a Hadamard manifold, and of a biharmonic Steklov problem on bounded domains of a Euclidean space. Especially, interesting rigidity results can be obtained if sharp bounds were achieved.
We prove a Sobolev inequality which holds on submanifolds in Euclidean space of arbitrary dimension and codimension. This inequality is sharp if the codimension is at most 2. As a special case, we obtain a sharp isoperimetric inequality for minimal submanifolds in Euclidean space of codimension at most 2.
Given a compact Riemannian manifold $M$ without boundary, we show that large isoperimetric regions in $M\times\mathbb{R}^k$ are tubular neighborhoods of $M\times\{x\}$, with $x\in\mathbb{R}^k$.
In this paper we consider the problem of minimizing the relative perimeter under a volume constraint in the interior of a convex body, i.e., a compact convex set in Euclidean space with interior points. We shall not impose any regularity assumption on the boundary of the convex set. Amongst other results, we shall prove the equivalence between Hausdorff and Lipschitz convergence, the continuity of the isoperimetric profile with respect to the Hausdorff distance,and the convergence in Hausdorff distance of sequences of isoperimetric regions and their free boundaries. We shall also describe the …
Isoperimetric regions minimize the size of their boundaries among all regions with the same volume. In Euclidean and Hyperbolic space, isoperimetric regions are round balls. We show that isoperimetric regions in two and three-dimensional nonpositively curved manifolds are not necessarily balls, and need not even be connected.
Let $Σ$ be a $k$-dimensional complete proper minimal submanifold in the Poincaré ball model $B^n$ of hyperbolic geometry. If we consider $Σ$ as a subset of the unit ball $B^n$ in Euclidean space, we can measure the Euclidean volumes of the given minimal submanifold $Σ$ and the ideal boundary $\partial_\infty Σ$, say $\rvol(Σ)$ and $\rvol(\partial_\infty Σ)$, respectively. Using this concept, we prove an optimal linear isoperimetric inequality. We also prove that if $\rvol(\partial_\infty Σ) \geq \rvol(\mathbb{S}^{k-1})$, then $Σ$ satisfies the classical isoperimetric inequality. By proving the…
The aim of this paper is to give not only an explicit upper bound of the total Q-curvature but also an induced isoperimetric deficit formula for the complete conformal metrics on $\mathbb R^n$, $n\ge 3$ with scalar curvature being nonnegative near infinity and Q-curvature being absolutely convergent.
We study the problem of existence of regions separating a given amount of volume with the least possible perimeter inside a Euclidean cone. Our main result shows that nonexistence for a given volume implies that the isoperimetric profile of the cone coincides with the one of the half-space. This allows us to give some criteria ensuring existence of isoperimetric regions: for instance, local convexity of the cone at some boundary point. We also characterize which are the stable regions in a convex cone, i.e., second order minima of perimeter under a volume constraint. From this it follows that …
In this paper we provide new existence results for isoperimetric sets of large volume in Riemannian manifolds with nonnegative Ricci curvature and Euclidean volume growth. We find sufficient conditions for their existence in terms of the geometry at infinity of the manifold. As a byproduct we show that isoperimetric sets of big volume always exist on manifolds with nonnegative sectional curvature and Euclidean volume growth. Our method combines an asymptotic mass decomposition result for minimizing sequences, a sharp isoperimetric inequality on nonsmooth spaces, and the concavity property of t…
For the biharmonic Steklov eigenvalue problem considered in this paper, we show that among all bounded Euclidean domains of class $C^{1}$ with fixed measure, the ball maximizes the first positive eigenvalue.
Motivated by Perelman's Pseudo Locality Theorem for the Ricci flow, we prove that if a Riemannian manifold has Ricci curvature bounded below in a metric ball which moreover has almost maximal volume, then in a smaller ball (in a quantified sense) it holds an almost-euclidean isoperimetric inequality. The result is actually established in the more general framework of non-smooth spaces satisfying local Ricci curvature lower bounds in a synthetic sense via optimal transportation.
We prove that the isoperimetric inequalities in the euclidean and hyperbolic plane hold for all euclidean, respectively hyperbolic, cone-metrics on a disk with singularities of negative curvature. This is a discrete analog of the theorems of Weil and Bol that deal with Riemannian metrics of curvature bounded from above by 0, respectively by -1. A stronger discrete version was proved by A.D.Alexandrov, with a subsequent extension by approximation to metrics of bounded integral curvature. Our proof uses "discrete conformal deformations" of the metric that eliminate the singularities and increase…
In this paper, the isoperimetric problem in Randers planes, $(\mathbb{R}^2,F=α+β)$, which are slight deformation of the Euclidean plane $(\mathbb{R}^2,α)$ by suitable one forms $β$, have been studied. We prove that the circles centred at the origin achieves the local maximum area of the isoperimetric problem with respect to well known volume forms in Finsler geometry.
In this note, an isoperimetric inequality for the harmonic mean of lower order Steklov eigenvalues is proved on bounded domains in noncompact rank-$1$ symmetric spaces. This work extends result of \cite{BR.01} and \cite{V.21} proved for bounded domains in Euclidean space and Hyperbolic space, respectively.
In this article, we prove an isoperimetric inequality for the harmonic mean of the first $(n-1)$ nonzero Steklov eigenvalues on bounded domains in $n$-dimensional Hyperbolic space. Our approach to prove this result also gives a similar inequality for the first $n$ nonzero Steklov eigenvalues on bounded domains in $n$-dimensional Euclidean space.
We study the isoperimetric problem in Euclidean space endowed with a density. We first consider piecewise constant densities and examine particular cases related to the characteristic functions of half-planes, strips and balls. We also consider continuous modification of Gauss density in $\R^2$. Finally, we give a list of related open questions.
We prove that if a quasiconvex subset $X$ of a metric space $Y$ has finite Nagata dimension and is Lipschitz $k$-connected or admits Euclidean isoperimetric inequalities up to dimension $k$ for some $k$ then $X$ is isoperimetrically undistorted in $Y$ up to dimension $k+1$. This generalizes and strengthens a recent result of the third named author and has several consequences and applications. It yields for example that in spaces of finite Nagata dimension, Lipschitz connectedness implies Euclidean isoperimetric inequalities, and Euclidean isoperimetric inequalities imply coning inequalities. …
We study the topology of (properly) immersed complete minimal surfaces $P^2$ in Hyperbolic and Euclidean spaces which have finite total extrinsic curvature, using some isoperimetric inequalities satisfied by the extrinsic balls in these surfaces, (see \cite{Pa}). We present an alternative and partially unified proof of the Chern-Osserman inequality satisfied by these minimal surfaces, (in $\erre^n$ and in $\Han$), based in the isoperimetric analysis above alluded. Finally, we show a Chern-Osserman type equality attained by complete minimal surfaces in the Hyperbolic space with finite total ext…
The aim of this paper is to prove isoperimetric inequalities on submanifolds of the Euclidean space using mass transportation methods. We obtain a sharp ?weighted isoperimetric inequality? and a nonsharp classical inequality similar to the one obtained by J. Michael and L. Simon. The proof relies on the description of a solution of the problem of Monge when the initial measure is supported in a submanifold and the final one supported in a linear subspace of the same dimension.
We study the problem of existence of isoperimetric regions for large volumes, in $C^0$-locally asymptotically Euclidean Riemannian manifolds with a finite number of $C^0$-asymptotically Schwarzschild ends. Then we give a geometric characterization of these isoperimetric regions, extending previous results contained in [EM13b], [EM13a], and [BE13]. Moreover strengthening a little bit the speed of convergence to the Schwarzschild metric we obtain existence of isoperimetric regions for all volumes for a class of manifolds that we named $C^0$-strongly asymptotic Schwarzschild, extending results of…
In a compact orbifold, for small prescribed volume, an isoperimetric region is close to a small metric ball; in a Euclidean orbifold, it is a small metric ball.
We prove that a proper geodesic metric space has non-positive curvature in the sense of Alexandrov if and only if it satisfies the Euclidean isoperimetric inequality for curves. Our result extends to non-geodesic spaces and non-zero curvature bounds.
We prove a pair of sharp reverse isoperimetric inequalities for domains in nonpositively curved surfaces: (1) metric disks centered at the vertex of a Euclidean cone of angle at least $2π$ have minimal area among all nonpositively curved disks of the same perimeter and the same total curvature; (2) geodesic triangles in a Euclidean (resp. hyperbolic) cone of angle at least $2π$ have minimal area among all nonpositively curved geodesic triangles (resp. all geodesic triangles of curvature at most $-1$) with the same side lengths and angles.
We establish sufficient conditions for existence of curves minimizing length as measured with respect to a degenerate metric on the plane while enclosing a specified amount of Euclidean area. Non-existence of minimizers can occur and examples are provided. This continues the investigation begun in [ABCDS] where the metric ds^2 near the singularities equals a quadratic polynomial times the standard metric. Here we allow the conformal factor to be any smooth non-negative potential vanishing at isolated points provided the Hessian at these points is positive definite. These isoperimetric curves, …
We show that the unique isoperimetric hypersurfaces in $\mathbb{R}^n$ with density $r^p$ for $n \ge 3$ and $p>0$ are spheres that pass through the origin.
We show that the Cheeger constant of compact surfaces is bounded by a function of the area. We apply this to isoperimetric profiles of bounded genus non-compact surfaces, to show that if their isoperimetric profile grows faster than $\sqrt t$, then it grows at least as fast as a linear function. This generalizes a result of Gromov for simply connected surfaces. We study the isoperimetric problem in dimension 3. We show that if the filling volume function in dimension 2 is Euclidean, while in dimension 3 is sub-Euclidean and there is a $g$ such that minimizers in dimension 3 have genus at most …
We establish a new symmetrization procedure for the isoperimetric problem in symmetric spaces of noncompact type. This symmetrization generalizes the well known Steiner symmetrization in euclidean space. In contrast to the classical construction the symmetrized domain is obtained by solving a nonlinear elliptic equation of mean curvature type. We conclude the paper discussing possible applications to the isoperimetric problem in symmetric spaces of noncompact type.
In this paper the author studies the isoperimetric problem in $\re^n$ with perimeter density $|x|^p$ and volume density $1.$ We settle completely the case $n=2,$ completing a previous work by the author: we characterize the case of equality if $0\leq p\leq 1$ and deal with the case $-\infty<p<-1$ (with the additional assumption $0\inΩ$). In the case $n\geq 3$ we deal mainly with the case $-\infty<p<0,$ showing among others that the results in $2$ dimensions do not generalize for the range $-n+1<p<0.$
For general varifolds in Euclidean space, we prove an isoperimetric inequality, adapt the basic theory of generalised weakly differentiable functions, and obtain several Sobolev type inequalities. We thereby intend to facilitate the use of varifold theory in the study of diffused surfaces.
We prove a generalized isoperimetric inequality for a domain diffeomorphic to a sphere that replaces filling volume with $k$-dilation. Suppose $U$ is an open set in $\mathbb{R}^n$ diffeomorphic to a Euclidean $n$-ball. We show that in dimensions at least 4 there is a map from a standard Euclidean ball of radius about $vol(\partial U)^{1/(n-1)}$ to $U$, with degree 1 on the boundary, and $(n-1)$-dilation bounded by some constant only depending on $n$. We also give an example in dimension 3 of an open set where no such map with small $(n-1)$-dilation can be found. The generalized isoperimetric i…
We consider the punctured plane with volume density $|x|^α$ and perimeter density $|x|^β$. We show that centred balls are uniquely isoperimetric for indices $(α,β)$ which satisfy the conditions $α-β+1>0$, $α\leq 2β$ and $α(β+1)\leqβ^2$ except in the case $α=β=0$ which corresponds to the classical isoperimetric inequality.
In this paper, we prove a class of weighted isoperimetric inequalities for bounded domains in hyperbolic space by using the isoperimetric inequality with log-convex density in Euclidean space. As a consequence, we remove the horo-convex assumption of domains in a weighted isoperimetric inequality proved by Scheuer-Xia. Furthermore, we prove weighted isoperimetric inequalities for star-shaped domains in warped product manifolds. Particularly, we obtain a weighted isoperimetric inequality for star-shaped hypersurfaces lying outside a certain radial coordinate slice in the anti-de Sitter-Schwarzs…
We study the Riemannian quantiative isoperimetric inequality. We show that direct analogue of the Euclidean quantitative isoperimetric inequality is--in general--false on a closed Riemannian manifold. In spite of this, we show that the inequality is true generically. Moreover, we show that a modified (but sharp) version of the quantitative isoperimetric inequality holds for a real analytic metric, using the Lojasiewicz-Simon inequality. A main novelty of our work is that in all our results we do not require any a priori knowledge on the structure/shape of the minimizers.
In this article we exhibit the largest constant in a quadratic isoperimetric inequality which ensures that a geodesic metric space is Gromov hyperbolic. As a particular consequence we obtain that Euclidean space is a borderline case for Gromov hyperbolicity in terms of the isoperimetric function. We prove similar results for the linear filling radius inequality. Our results strengthen and generalize theorems of Gromov, Papasoglu and others.
We consider the isoperimetric problem in planar sectors with density $r^{p}$, and with density $a>1$ inside the unit disk and $1$ outside. We characterize solutions as a function of sector angle. We also solve the isoperimetric problem in $\mathbb{R}^{n}$ with density $r^{p},\; p<0$.
We extend several Cheeger-type isoperimetric bounds for convex sets in Euclidean space, due to Bobkov and Kannan-Lovász-Simonovits, to Riemannian manifolds having non-negative Ricci curvature. In order to extend Bobkov's bound, we require in addition an upper bound on the sectional curvature of the space, which permits us to use comparison tools in Cartan-Alexandrov-Toponogov (or CAT) spaces. Along the way, we also quantitatively improve our previous result that weak concentration assumptions imply a Cheeger-type isoperimetric bound, to a sharp bound with respect to all parameters.
We study stable smooth solutions to the isoperimetric type problem for a Gaussian weight on Euclidean Space. That is, we study hypersurfaces $Σ^n \subset \mathbb R^{n+1}$ that are second order stable critical points of compact variations that minimize Gaussian weighted area and preserve Gaussian weighted volume. We show that such $Σ$ satisfy a curvature condition, and derive the Jacobi operator $L$ for the second variation of such $Σ$. Our first main result is that for non-planar $Σ$, bounds on the index of $L$, acting on volume preserving variations, gives us that $Σ$ splits off a linear spac…
Given a positive lower semi-continuous density $f$ on $\mathbb{R}^2$ the weighted volume $V_f:=f\mathscr{L}^2$ is defined on the $\mathscr{L}^2$-measurable sets in $\mathbb{R}^2$. The $f$-weighted perimeter of a set of finite perimeter $E$ in $\mathbb{R}^2$ is written $P_f(E)$. We study minimisers for the weighted isoperimetric problem \[ I_f(v):=\inf\Big\{ P_f(E):E\text{ is a set of finite perimeter in }\mathbb{R}^2\text{ and }V_f(E)=v\Big\} \] for $v>0$. Suppose $f$ takes the form $f:\mathbb{R}^2\rightarrow(0,+\infty);x\mapsto e^{h(|x|)}$ where $h:[0,+\infty)\rightarrow\mathbb{R}$ is a non-d…
Given a compact Riemannian manifold with density $M$ without boundary and the real line $\mathbb{R}$ with constant density, we prove that isoperimetric regions of large volume in $M\times\mathbb{R}$ with the product density are slabs of the form $M\times [a,b]$. We previously prove, as a necessary step, the existence of isoperimetric regions in any manifold of density where a subgroup of the group of transformations preserving weighted perimeter and volume acts cocompactly.
We completely characterize isoperimetric regions in R^n with density e^h, where h is convex, smooth, and radially symmetric. In particular, balls around the origin constitute isoperimetric regions of any given volume, proving the Log-Convex Density Conjecture due to Kenneth Brakke.
We prove some old and new isoperimetric inequalities with the best constant using the ABP method applied to an appropriate linear Neumann problem. More precisely, we obtain a new family of sharp isoperimetric inequalities with weights (also called densities) in open convex cones of $\mathbb{R}^n$. Our result applies to all nonnegative homogeneous weights satisfying a concavity condition in the cone. Remarkably, Euclidean balls centered at the origin (intersected with the cone) minimize the weighted isoperimetric quotient, even if all our weights are nonradial ---except for the constant ones. W…
