arXiv research

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

Study on isoperimetric problem in Randers planes achieving maximum area.

Study minimizers in large volume isoperimetric problems with a new flatness criterion.

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…

Study finds optimal loops in hyperbolic space with Finsler structure.

Survey on soap bubble partitions and their stability.

Survey on curvature bounds and isoperimetric inequalities.

In the main theorem of this paper we treat the problem of existence of minimizers of the isoperimetric problem under the assumption of small volumes. Applications of the main theorem to asymptotic expansions of the isoperimetric problem are given.

The paper studies nonlocal isoperimetric problems in hyperbolic space and finds unique minimizers for small volumes.

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 $r^{p},\; p<0$.

Study the isoperimetric problem in Riemannian manifolds with non-trivial conformal vector fields.

Solves relative isoperimetric problem for cubes, identifying specific minimizers.

The article proves Randers Poincaré disc satisfies isoperimetric equality.

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 the isoperimetric problem.

Study proves quantitative results for isoperimetric problem outside convex bodies in the plane.

Overview of recent results on isoperimetric inequalities on manifolds with Ricci lower bounds.

We provide a partial solution to the isoperimetric problem in the Heisenberg group.

We consider three generalizations of the isoperimetric problem to higher codimension and provide results on equilibrium, stability, and minimization.

New proof confirms flat equilateral torus is λ1-maximal.

Solves isoperimetric problem for curl operator on compact 3-manifolds.

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…

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

Local minimizers are convex and close to Wulff shapes.

This paper is a continuation of the second author's previous work. We investigate the isoperimetric problem in the 2-dimensional Finsler space form $(F_B, B^2(1))$ with $k=0$ by using the Holmes-Thompson area and prove that the circle centered the origin achieves the local maximum area of the isoperimetric problem.

The paper solves reverse isoperimetric problems for convex bodies with curvature constraints.

The study solves the isoperimetric problem for Heisenberg group norms.

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…

Geodesic balls are isoperimetric in hyperbolic spaces with certain densities.

Proves strict inequality for minimizers of Willmore energy under isoperimetric constraints.

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 equatio…

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

Study on minimal disks in metric spaces, focusing on branch set structure.

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…

In this paper we consider the problem of minimizing the relative perimeter under a volume constraint in the interior of a conically bounded convex set, i.e., an unbounded convex body admitting an \emph{exterior} asymptotic cone. Results concerning existence of isoperimetric regions, the behavior of the isoperimetric pr…

Researchers find a surface with minimum bending energy for any genus and isoperimetric ratio.

The paper proves isoperimetric regions on Riemannian manifolds with Ricci bounded below.

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…

Sharp inequality outside ball proved using Neumann method.

Locally isoperimetric partitions minimize perimeter in space.

Square Clifford torus uniquely determined by isoperimetric ratio, rectangular torus not.

Study nonnegatively curved Alexandrov spaces, proving isoperimetric conditions and structure at infinity.

We consider the class of $λ$-concave bodies in $\mathbb R^{n+1}$; that is, convex bodies with the property that each of their boundary points supports a tangent ball of radius $1/λ$ that lies locally (around the boundary point) inside the body. In this class we solve a reverse isoperimetric problem: we show that the co…

In this work we investigate the following isoperimetric problem: to find the regions of prescribed volume with minimal boundary area between two parallel horospheres in hyperbolic 3-space (the area of the part of the boundary contained in the horospheres is not included). We reduce the problem to the study of rotationa…

The paper solves the isoperimetric problem on noncompact metric spaces with lower Ricci bounds.

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 …

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 prov…

New proof of Wulff-Gage inequality with applications.

In this paper we study the relationship of hyperbolicity and (Cheeger) isoperimetric inequality in the context of Riemannian manifolds and graphs. We characterize the hyperbolic manifolds and graphs (with bounded local geometry) verifying this isoperimetric inequality, in terms of their Gromov boundary. Furthermore, we…