arXiv research

We prove isoperimetric inequalities for quotients of $n$-dimensional Affine buildings. We use these inequalities to prove topological overlapping for the 2-dimensional skeletons of these buildings.

Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n$ with smooth boundary $\partial M$. Suppose that $(M,g)$ admits a scalar-flat conformal metric. We prove that the supremum of the isoperimetric quotient over the scalar-flat conformal class is strictly larger than the best constant of the isoperimetric…

The renormalized volume is reinterpreted using isoperimetric profiles.

In this work the Isoperimetric Inequality for integral varifolds is used to obtain sharp estimates for the size of the set where the density quotient is small and to generalise Calderón's and Zygmund's theory of first order differentiability for functions in Lebesgue spaces from Lebesgue measure to integral varifolds.

In this paper we provide an extension to the Jellett-Minkowski's formula for immersed submanifolds into ambient manifolds which possesses a pole and radial curvatures bounded from above or below by the radial sectional curvatures of a rotationally symmetric model space. Using this Jellett-Minkowski's generalized formul…

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…

Let $(M,g)$ be a complete non-compact Riemannian manifold together with a function $e^h$, which weights the Hausdorff measures associated to the Riemannian metric. In this work we assume lower or upper radial bounds on some weighted or unweighted curvatures of $M$ to deduce comparisons for the weighted isoperimetric qu…

We prove existence of regions minimizing perimeter under a volume constraint in contact sub-Riemannian manifolds such that their quotient by the group of contact transformations preserving the sub-Riemannian metric is compact.

Study mass and center of mass in flat 3-manifolds, proving existence of foliations.

The paper studies isoperimetric inequalities on warped product manifolds.

The aim of this article is: (a) To establish the existence of the best isoperimetric constants for the $(H^1,BMO)$-normal conformal metrics $e^{2u}|dx|^2$ on $\mathbb R^n$, $n\ge 3$, i.e., the conformal metrics with the Q-curvature orientated conditions $$ (-Δ)^{n/2}u\in H^1(\mathbb R^n) & \ u(x)=\hbox{const.}+\frac{\i…

Study non-local isoperimetric energies on spheres using a Riemannian autocorrelation function.

Develops comparison methods for semilinear elliptic problems on Riemannian manifolds with Ricci lower bound.

We obtain upper bounds for the isoperimetric quotients of extrinsic balls of submanifolds in ambient spaces which have a lower bound on their radial sectional curvatures. The submanifolds are themselves only assumed to have lower bounds on the radial part of the mean curvature vector field and on the radial part of the…

Groups satisfy linear surface isoperimetric functions.

Solves relative isoperimetric problem on polygonal domains, focusing on corners.

The isoperimetric inequality and related inequalities are explored.

Isoperimetric regions in scaled product manifolds are products of regions in each factor.

The paper finds new inequalities for convex polygons.

Huisken's isoperimetric mass is always nonnegative.

In curved spaces, isoperimetric sets don't exist for small volumes.

Simplified proof for Cheeger's isoperimetric constant.

The study proves the existence and properties of isoperimetric clusters in Riemannian manifolds with bounded geometry.

Study improves isoperimetric inequality for random groups.

The paper proves conditions for isoperimetric regions in curved spaces.

The paper proves new inequalities in hyperbolic space using Euclidean methods.

The study proves analogues of the discrete isoperimetric inequality in hyperbolic geometry.

We consider the problem of minimizing the relative perimeter under a volume constraint in an unbounded convex body $C\subset \mathbb{R}^{n+1}$, without assuming any further regularity on the boundary of $C$. Motivated by an example of an unbounded convex body with null isoperimetric profile, we introduce the concept of…

Proves optimal isoperimetric inequality in de Sitter space.

Sharp isoperimetric inequality for Finsler manifolds with non-negative Ricci curvature.

Generic smooth boundaries for isoperimetric regions in 8D manifolds.

The paper characterizes hyperbolic manifolds and graphs verifying a specific isoperimetric inequality.

Proof of reverse isoperimetric inequality for black holes.

The study finds a special isoperimetric inequality for minimal hypersurfaces in spheres.

Solves equality case in isoperimetric inequality for non-convex domains.

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 con…

Study connects isoperimetric sets, mass concepts, and nonnegative scalar curvature.

Prove an isoperimetric inequality for compact bodies in 3D contact non-unimodular Lie groups.

Local isoperimetric inequality holds for balls with nonpositive curvature.

New isoperimetric inequalities in the plane with radial weights identified.

New findings on metric spaces with finite Nagata dimension.

Study on isoperimetric problem in Randers planes achieving maximum area.

We describe explicitly the large volume isoperimetric regions of a natural class of asymptotically flat manifolds, in any dimension. These isoperimetric regions detect the mass and the center of mass of such manifolds when viewed as initial data sets for the Einstein equations in general relativity. Using the positivit…

The paper finds singular isoperimetric regions in high-dimensional spaces.

We show that any star-shaped convex hypersurface with constant Weingarten curvature in the deSitter-Schwarzschild manifold is a sphere of symmetry. Moreover, we study an isoperimetric problem for bounded domains in the doubled Schwarzschild manifold. We prove the existence of an isoperimetric surface for any value of t…

We solve the isoperimetric problem in the Lens spaces with large fundamental group. Namely, we prove that the isoperimetric surfaces are geodesic spheres or tori of revolution about geodesics. We also show that the isoperimetric problem in L(3,1) and L(3,2) follows from the proof of the Willmore conjecture by Marques a…

Based on Markvorsen and Palmer's work on mean time exit and isoperimetric inequalities we establish slightly better isoperimetric inequalities and mean time exit estimates for minimal submanifolds of $N\times\mathbb{R}$. We also prove isoperimetric inequalities for submanifolds of Hadamard spaces with tamed second fund…

Smooth minimizers found for Willmore energy surfaces.