arXiv research

Study compares isoperimetric profiles on manifolds with integral Ricci curvature bounds.

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…

A comparison theorem for the isoperimetric profile on the universal cover of surfaces evolving by normalised Ricci flow is proven. For any initial metric, a model comparison is constructed that initially lies below the profile of the initial metric and which converges to the profile of the constant curvature metric. Th…

Paper proves a sharp weighted Isoperimetric inequality for substatic manifolds.

The paper develops new methods to study sharp isoperimetric properties on complex spaces.

Study geometric and topological properties of Finsler manifolds with weighted Ricci curvature bounds.

We prove a comparison theorem for the isoperimetric profiles of simple closed curves evolving by the normalized curve shortening flow: If the isoperimetric profile of the region enclosed by the initial curve is greater than that of some `model' convex region with exactly four vertices and with reflection symmetry in bo…

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

We prove a comparison theorem for the isoperimetric profiles of solutions of the normalized Ricci flow on the two-sphere: If the isoperimetric profile of the initial metric is greater than that of some positively curved axisymmetric metric, then the inequality remains true for the isoperimetric profiles of the evolved …

Bishop's volume comparison theorem states that a compact $n$-manifold with Ricci curvature larger than the standard $n$-sphere has less volume. While the traditional proof uses geodesic balls, we present another proof using isoperimetric hypersurfaces, also known as "soap bubbles," which minimize area for a given volum…

We prove that the isoperimetric profile of a convex domain $Ω$ with compact closure in a Riemannian manifold $(M^{n+1},g)$ satisfies a second order differential inequality which only depends on the dimension of the manifold and on a lower bound on the Ricci curvature of $Ω$. Regularity properties of the profile and top…

Paper compares solutions of Poisson equations on Riemannian manifolds with Robin boundary.

We provide an isoperimetric comparison theorem for small volumes in an $n$-dimensional Riemannian manifold $(M^n,g)$ with strong bounded geometry, as in Definition $2.3$, involving the scalar curvature function. Namely in strong bounded geometry, if the supremum of scalar curvature function $S_g<n(n-1)k_0$ for some $k_…

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

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

We extend several Cheeger-type isoperimetric bounds for convex sets in Euclidean space, due to Bobkov and Kannan-Lovász-Simonovits, to Riemannian manifolds having non-negative Ricci curvature. In order to extend Bobkov's bound, we require in addition an upper bound on the sectional curvature of the space, which permits…

We estimate from below the isoperimetric profile of $S^2 \times \re^2$ and use this information to obtain lower bounds for the Yamabe constant of $S^2 \times \re^2$. This provides a lower bound for the Yamabe invariants of products $S^2 \times M^2$ for any closed Riemann surface $M$. Explicitly we show that $Y(S^2 \tim…

We prove spectral, stochastic and mean curvature estimates for complete $m$-submanifolds $\varphi \colon M \to N$ of $n$-manifolds with a pole $N$ in terms of the comparison isoperimetric ratio $I_{m}$ and the extrinsic radius $r_\varphi\leq \infty$. Our proof holds for the bounded case $r_\varphi< \infty$, recovering …

In this work we consider a question in the calculus of variations motivated by riemannian geometry, the isoperimetric problem. We show that solutions to the isoperimetric problem, close in the flat norm to a smooth submanifold, are themselves smooth and $C^{2,α}$-close to the given sub manifold. We show also a version …

Study shows eigenvalue of Hodge Laplacian on coexact 1-forms in hyperbolic 3-manifolds is related to isoperimetric ratio.

Proves Riemannian Penrose Inequality for specific manifolds.

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…

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

Paper proves isoperimetric inequality for Minkowski spacetime.

We prove that if a geodesic metric measure space satisfies a comparison condition for isoperimetric profile and if the observable variance is maximal, then the space is foliated by minimal geodesics, where the observable variance is defined to be the supremum of the variance of 1-Lipschitz functions on the space. Our r…

We obtain new sharp isoperimetric inequalities on a Riemannian manifold equipped with a probability measure, whose generalized Ricci curvature is bounded from below (possibly negatively), and generalized dimension and diameter of the convex support are bounded from above (possibly infinitely). Our inequalities are shar…

Sharp estimates for parabolic equations on manifolds using symmetrization.

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…

We prove that on a Riemannian manifold, a smooth differential form has a primitive with a given (functional) upper bound provided the necessary weighted isoperimetric inequalities implied by Stokes are satisfied. We apply this to prove a comparison predicted by Gromov between the cofilling function and the filling area…

The paper proves isoperimetric inequalities in manifolds with small negative Ricci curvature.

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

We study the isoperimetric structure of asymptotically flat Riemannian 3-manifolds (M,g) that are C^0-asymptotic to Schwarzschild of mass m>0. Refining an argument due to H. Bray we obtain an effective volume comparison theorem in Schwarzschild. We use it to show that isoperimetric regions exist in (M, g) for all suffi…

This is a survey article on recent progress of comparison geometry and geometric analysis on Finsler manifolds of weighted Ricci curvature bounded below. Our purpose is two-fold: Give a concise and geometric review on the birth of weighted Ricci curvature and its applications; Explain recent results from a nonlinear an…

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 inequalities on hyperbolic spaces and Riemannian manifolds using symmetrization and heat semigroup.

We present a new proof of the sphere covering inequality in the spirit of comparison geometry, and as a byproduct we find another sphere covering inequality which can be viewed as the dual of the original one. We also prove sphere covering inequalities on surfaces satisfying general isoperimetric inequalities, and disc…

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.

A new isoperimetric estimate is proved for embedded closed curves evolving by curve shortening flow, normalized to have total length $2π$. The estimate bounds the length of any chord from below in terms of the arc length between its endpoints and elapsed time. Applying the estimate to short segments we deduce directly …

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.

We study the biharmonic Steklov eigenvalue problem on a compact Riemannian manifold $Ω$ with smooth boundary. We give a computable, sharp lower bound of the first eigenvalue of this problem, which depends only on the dimension, a lower bound of the Ricci curvature of the domain, a lower bound of the mean curvature of i…

Study improves isoperimetric inequality for random groups.