arXiv research

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

Improved bounds on curve filling areas in Banach spaces, leading to rigidity of Pu's inequality.

Study proves topological properties of isoperimetric sets in specific spaces.

Critical metrics in Levy-Gromov inequality are rigid in 2D.

The paper proves local rigidity theorems for scalar curvature and related inequalities.

Optimal constants for isoperimetric inequalities involving Steklov eigenvalues on surfaces are determined.

We characterize the standard $\mathbb{S}^3$ as the closed Ricci-positive 3-manifold with scalar curvature at least 6 having isoperimetric surfaces of largest area: $4π$. As a corollary we answer in the affirmative an interesting special case of a conjecture of Min-Oo's on the scalar curvature rigidity of the upper hemi…

Boundary rigidity defined for hyperbolic spaces, tied to geometric properties.

The paper finds bounds for eigenvalues of various operators on domains and hypersurfaces.

In this note, we consider the isoperimetric inequality on an asymptotically flat manifold with nonnegative scalar curvature, and improve it by using Hawking mass. We also obtain a rigidity result when equality holds for the classical isoperimetric inequality on an asymptotically flat manifold with nonnegative scalar cu…

Sharp isoperimetric inequality proven for specific metric measure spaces.

Study on $p$-Laplace equation in convex cones, proving rigidity under specific conditions.

New theorem splits spaces with maximal variance of 1-Lipschitz functions.

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…

New isoperimetric inequality for clamped plates in RCD(0,N) spaces, sharp and stable.

We establish some a priori geometric relations on stable minimal surfaces lying inside three-manifolds with scalar curvature uniformly bounded below. The relations are based on a slight generalization of a formula due to Castillon. We apply it to prove non-local rigidity results in the particular sense that they expres…

We prove that if $(X,\mathsf{d},\mathfrak{m})$ is a metric measure space with $\mathfrak{m}(X)=1$ having (in a synthetic sense) Ricci curvature bounded from below by $K>0$ and dimension bounded above by $N\in [1,\infty)$, then the classic Lévy-Gromov isoperimetric inequality (together with the recent sharpening counter…

New method uses entropy dissipation to prove isoperimetric inequalities.

The paper explores sharp isoperimetric properties on non-compact spaces with Ricci bounds.

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

Study on isoperimetric inequality on weighted Riemannian manifolds with negative effective dimension.

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

The paper proves stability for Möbius transformations in high dimensions.

Study rigidity of spectral gap on Finsler manifolds with specific curvature bounds.

The paper proves curvature rigidity for manifolds with specific scalar curvature bounds.

Sharp inequalities in nonnegative Ricci curvature spaces using mass transport.

Study on rigidity of logarithmic Sobolev inequality on manifolds.

Analyzes metric spaces homeomorphic to manifolds, proving rigidity and inequalities.

Study stability of rigid motions and Möbius transformations on spheres, proving new rigidity estimates.

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

Let $\,(M,g)\,$ be a $n$-dimensional Riemannian manifold and $\,Ω\,$ be any compact connected domain in $\,M$. We study the problem of finding the {\em maxima} of the functional $\, {\mathcal E} (Ω)\,$ (known as {\em torsional rigidity} associated to $Ω$) among all domains of prescribed volume $v$. Our results show tha…

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

We investigate the minimal and isoperimetric surface problems in a large class of sub-Riemannian manifolds, the so-called Vertically Rigid spaces. We construct an adapted connection for such spaces and, using the variational tools of Bryant, Griffiths and Grossman, derive succinct forms of the Euler-Lagrange equations …

The study characterizes convex bodies with equal isoperimetric profiles to half-spaces and estimates their volume behavior.

Sharp inequality for Lorentzian spaces with timelike Ricci bounds.

The article proves a Poincaré inequality for hypersurfaces and applies it to rigidity results.

In this article we prove a reverse Hölder inequality for the fundamental eigenfunction of the Dirichlet problem on domains of a compact Riemannian manifold with lower Ricci curvature bounds. We also prove an isoperimetric inequality for the torsional ridigity of such domains.

We prove explicit upper and lower bounds for the torsional rigidity of extrinsic domains of submanifolds P^m with controlled radial mean curvature in ambient Riemannian manifolds N^n with a pole p and with sectional curvatures bounded from above and from below, respectively. These bounds are given in terms of the torsi…

The paper extends rigidity results for $λ$-self-expanders to hyperplanes, spheres, and cylinders.

The paper studies billiards in symmetric tables and finds a measure bound for maximizing orbits.

The paper proves partial rigidity of Hawking mass for stable CMC spheres in specific manifolds.

Sharp spectral theorems and isoperimetric inequalities for manifolds with nonnegative Ricci curvature.

Study surfaces with nonnegative curvature in spectral sense, proving inequalities and bounds.

The paper proves entropy power properties on Riemannian manifolds and Ricci flows.

The paper extends Liouville's theorem to calibrated geometries in various dimensions.

In this paper, firstly, inspired by Natário's recent work \cite{Na}, we use the isoperimetric inequality to derive some Alexandrov-Fenchel type inequalities for closed convex hypersurfaces in the hyperbolic space $\H^{n+1}$ and in the sphere $\SS^{n+1}$. We also get the rigidity in the spherical case. Secondly, we use …

Study fine Pólya-Szegő inequalities in metric spaces with applications.

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