arXiv research

Viscosity solutions are suitable notions in the study of nonlinear PDEs justified by estimates established via the maximum principle or the comparison principle. Here we prove that the isoperimetric profile functions of Riemannian manifolds with Ricci lower bound are viscosity super-solutions of some nonlinear differen…

Study sharp inequalities for perimeter functionals in capillarity and convex cones.

New method uses entropy dissipation to prove isoperimetric inequalities.

Both analytic and geometric forms of an optimal monotone principle for $L^p$-integral of the Green function of a simply-connected planar domain $Ω$ with rectifiable simple curve as boundary are established through a sharp one-dimensional power integral estimate of Riemann-Stieltjes type and the Huber analytic and geome…

The paper analyzes the emergence of almost-honeycomb structures in low-energy planar clusters.

We adapt the theory of currents in metric spaces, as developed by the first-mentioned author in collaboration with B. Kirchheim, to currents with coefficients in Z_p. Building on S. Wenger's work in the orientable case, we obtain isoperimetric inequalities mod(p) in Banach spaces and we apply these inequalities to prov…

Study on lengths and curvatures of harmonic functions on smooth and singular surfaces.

In this paper we consider the polyharmonic heat flow of a closed curve in the plane. Our main result is that closed initial data with initially small normalised oscillation of curvature and isoperimetric defect flows exponentially fast in the C^infty-topology to a simple circle. Our results yield a characterisation of …

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.

We consider a convex Euclidean hypersurface that evolves by a volume or area preserving flow with speed given by a general nonhomogeneous function of the mean curvature. For a broad class of possible speed functions, we show that any closed convex hypersurface converges to a round sphere. The proof is based on the mono…

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.

The study proves inequalities and curvature properties for Markov chains.

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.

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

Study proves topological properties of isoperimetric sets in specific spaces.

For a complete noncompact connected Riemannian manifold with bounded geometry, we prove the existence of isoperimetric regions in a larger space obtained by adding finitely many limit manifolds at infinity. As one of many possible applications, we extend properties of the isoperimetric profile from compact manifolds to…

Constructs isoperimetric regions from separating hypersurfaces.