arXiv research

We consider sets of locally finite perimeter in Carnot groups. We show that if E is a set of locally finite perimeter in a Carnot group G, then for almost every x in G with respect to the perimeter measure of E, some tangent of E at x is a vertical halfspace. This is a partial extension of a theorem of Franchi-Serapion…

The paper studies properties of $\mathrm{RCD}(K,N)$ spaces and their boundaries.

Minimal partitions with minimal perimeter found in metric spaces.

Infinite hyperbolic manifolds share same perimeter-to-volume ratio.

We show that among sets of finite perimeter balls are the only volume-constrained critical points of the perimeter functional.

Given an elliptic integrand of class $\mathscr{C}^{3}$, we prove that finite unions of disjoint open Wulff shapes with equal radii are the only volume-constrained critical points of the anisotropic surface energy among all sets with finite perimeter and reduced boundary almost equal to its closure.

This note is devoted to the study of sets of finite perimeter over RCD$(K,N)$ metric measure spaces. Its aim is to complete the picture about the generalization of De Giorgi's theorem within this framework. Starting from the results of [2] we obtain uniqueness of tangents and rectifiability for the reduced boundary of …

Paper proves isoperimetric inequality for Minkowski spacetime.

A lens cluster minimizes perimeter in the plane with given area constraints.

Study proves rigidity of critical points in hydrophobic capillary systems.

Study minimizers of quasi-perimeters in RCD spaces with volume constraints.

In the setting of Carnot groups, we are concerned with the rectifiability problem for subsets that have finite sub-Riemannian perimeter. We introduce a new notion of rectifiability that is, possibly, weaker than the one introduced by Franchi, Serapioni, and Serra Cassano. Namely, we consider subsets $Γ$ that, similarly…

We prove that the results regarding the Isoperimetric inequality and Cheeger constant formulated in terms of the Minkowski content, obtained by the authors in previous papers in the framework of essentially non-branching metric measure spaces verifying the local curvature dimension condition, also hold in the stronger …

Study fractional perimeter asymptotics on Riemannian manifolds as $s$ approaches 0.

Generic level sets in mean curvature flow are BV solutions.

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 isop…

We give a notion of BV function on an oriented manifold where a volume form and a family of lower semicontinuous quadratic forms $G_p: T_pM \to [0,\infty]$ are given. When we consider sub-Riemannian manifolds, our definition coincide with the one given in the more general context of metric measure spaces which are doub…

The paper classifies energy-minimizing sets in specific domains.

Study shows limits of volume-constrained sets are finite unions of Wulff shapes.

This paper characterizes semigenerated Carnot groups and applies it to rectifiability of perimeter sets.

A group is coherent if all its finitely generated subgroups are finitely presented. In this article we provide a criterion for positively determining the coherence of a group. This criterion is based upon the notion of the perimeter of a map between two finite 2-complexes which is introduced here. In the groups to whic…

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 $D\subset E\subset C$ has perimeter larger than or equal to the one of the rotationally symm…

Hexagonal tilings minimize perimeter with unequal volumes.

Paper finds formulas for minimizing perimeter in special spaces, proving key dimensions and existence.

Study on droplet flow on uneven surfaces, proving existence and properties.

The paper examines perimeter minimizing sets in curved spaces and finds conditions for their boundary to match a specific structure.

We show that all groups in a very large class of Coxeter groups are locally quasiconvex and have uniform membership problem solvable in quadratic time. If a group in the class satisfies a further hypothesis it is subgroup separable and relevant homomorphisms are also calculable in quadratic time. The algorithm also dec…

Proof shows smooth minimal hypersurfaces for perimeter-minimizing sets in low-dimensional Riemannian manifolds.

Flow preserves volume on flat torus, converging to stable set.

Perimeter on manifolds leads to new symmetrization methods.

Locally isoperimetric partitions minimize perimeter in space.

Perimeter minimizers in curved spaces have a singular set no more than 5 dimensions.

The generalized soap bubble problem seeks the least perimeter way to enclose and separate n given volumes in R^m. We study the possible configurations for perimeter minimizing bubble complexes enclosing more than two regions. We prove that perimeter minimizing planar bubble complexes with equal pressure regions and wit…

Geometric proof shows regularity of anisotropic minimal surfaces in 2D.

Geodesic spheres in certain symmetric spaces are quantitatively stable under small perturbations.

This study proves the Half Space Property for RCD(0,N) and RCD(K,N) spaces.

We describe all families of star-shaped n-polygons in the Euclidean plane with prescribed perimeter and area ; they are leaves of a foliation F on the space of star-shaped n-polygons. By the way, we study some geometric properties of convex polygons, for instance their inscriptibility in a circle and their regularity i…

We prove that the least-perimeter partition of the sphere into four regions of equal area is a tetrahedral partition.

We prove a monotonicity result at specific points for the Horizontal Perimeter for a class of surfaces in the Heisenberg group.

Almost perimeter-minimizing boundaries in plentiful groups can be approximated by Lipschitz graphs.

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

We prove the existence of a perimeter-minimizing partition of R^n into regions of unit volume. We conclude with a short tribute to the late Manuel A. Fortes.

We prove that a family of entire intrinsic minimal graphs in the Heisenberg group are not perimeter minimizing.

DUPLE tackles cross-deployment recognition in fiber-optic perimeter security with meta-learning.

We prove that finite perimeter subsets of $\mathbb{R}^{n+1}$ with small isoperimetric deficit have boundary Hausdorff-close to a sphere up to a subset of small measure. We also refine this closeness under some additional a priori integral curvature bounds. As an application, we answer a question raised by B. Colbois co…

The paper explores connections between perimeter, area, and visual angle of convex sets.

We prove that, among all convex hyperbolic polygons with given angles, the perimeter is minimized by the unique polygon with an inscribed circle. The proof relies on work of J.-M.\ Schlenker.

Two natural foliations, guided by area and perimeter, of the configurations spaces of planar polygons are considered and the topology of their leaves is investigated in some detail. In particular, the homology groups and the homotopy type of leaves are determined. The homology groups of the spaces of polygons with fixe…