arXiv research

Simplified proof for Cheeger's isoperimetric constant.

The Cheeger constant increases under Ricci flow on spheres.

Negative curvature manifolds have vanishing bounded volume class if and only if Cheeger constant is positive.

We obtain an estimate of the Cheeger isoperimetric constant in terms of the volume growth for a properly immersed submanifold in a Riemannian manifold which possesses at least one pole and sectional curvature bounded from above .

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

We use the concept of intrinsic metrics to give a new definition for an isoperimetric constant of a graph. We use this novel isoperimetric constant to prove a Cheeger-type estimate for the bottom of the spectrum which is nontrivial even if the vertex degrees are unbounded.

Study on Cheeger constant in specific hyperbolic 3-manifolds.

We give a brief literature review of the isoperimetric problem and discuss its relationship with the Cheeger constant of Riemannian $n$-manifolds. For some non-compact, finite area 2-manifolds, we prove the existence and regularity of subsets whose isoperimetric ratio is equal to the Cheeger constant. To do this, we us…

We prove a Cheeger inequality for the first positive Steklov eigenvalue. It involves two isoperimetric constants.

We show that the Cheeger isoperimetric constant of a solvable simply connected Lie group $G$ with Lie algebra $\G$ is $h(G)=\max_{H\in\G,||H||=1} \tr(\ad (H))$.

New proofs of Cheeger-like inequalities for coexact 1-forms on hyperbolic manifolds.

Study shows convergence rates for Cheeger cuts on data clouds.

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…

We show that the Cheeger constant of compact surfaces is bounded by a function of the area. We apply this to isoperimetric profiles of bounded genus non-compact surfaces, to show that if their isoperimetric profile grows faster than $\sqrt t$, then it grows at least as fast as a linear function. This generalizes a resu…

The study finds a special isoperimetric inequality for minimal hypersurfaces in spheres.

We show how 'test' vector fields may be used to give lower bounds for the Cheeger constant of a Euclidean domain (or Riemannian manifold with boundary), and hence for the lowest eigenvalue of the Dirichlet Laplacian on the domain. Also, we show that a continuous version of the classical Max Flow Min Cut Theorem for net…

We investigate the distribution of eigenvalues of the weighted Laplacian on closed weighted Riemannian manifolds of nonnegative Bakry-Émery Ricci curvature. We derive some universal inequalities among eigenvalues of the weighted Laplacian on such manifolds. These inequalities are quantitative versions of the previous t…

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 …

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

Sharp stability results for reverse isoperimetric inequalities in 2D.

The paper characterizes hyperbolic manifolds and graphs verifying a specific isoperimetric inequality.

We prove a lower bound for the $k$-th Steklov eigenvalues in terms of an isoperimetric constant called the $k$-th Cheeger-Steklov constant in three different situations: finite spaces, measurable spaces, and Riemannian manifolds. These lower bounds can be considered as higher order Cheeger type inequalities for the Ste…

Quantifies fractional isoperimetric inequality with strong control over boundary oscillation.

3D spheres with certain properties approach the round sphere.

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…

Given a non-compact, simply connected homogeneous three-manifold $X$ and a sequence $\{Ω_n\}_n$ of isoperimetric domains in $X$ with volumes tending to infinity, we prove that as $n\to \infty$: 1. The radii of the $Ω_n$ tend to infinity. 2. The ratios $\{Area} (\partial Ω_n)/\{Vol}(Ω_n)$ converge to the Cheeger consta…

The goal of the paper is to sharpen and generalise bounds involving the Cheeger's isoperimetric constant $h$ and the first eigenvalue $λ_{1}$ of the Laplacian. A celebrated lower bound of $λ_{1}$ in terms of $h$, $λ_{1}\geq h^{2}/4$, was proved by Cheeger in 1970 for smooth Riemannian manifolds. An upper bound on $λ_{1…

The paper studies isoperimetric inequalities on warped product manifolds.

We discuss the behavior of $(λ_{1. p}(M))^{1/p}$ with respect to the Gromov-Hausdorff topology and the variable $p$, where $λ_{1, p}(M)$ is the first positive eigenvalue of the $p$-Laplacian on a compact Riemannian manifold $M$. Applications include new estimates for the first eigenvalues of the $p$-Laplacian on Rieman…

The paper proves a diastolic inequality linking surface area and loop length.

New upper bound for Cheeger constant of hyperbolic surfaces.

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…

Random hyperbolic surfaces have low Cheeger constants.

The purpose of this paper is to analyze the isoperimetric inequality for symmetric log-convex probability measures on the line. Using geometric arguments we first re-prove that extremal sets in the isoperimetric inequality are intervals or complement of intervals (a result due to Bobkov and Houdré). Then we give a quan…

Study Cheeger inequalities for Riemannian manifolds with boundary.

Paper connects probability density cuts to graph theory eigenfunctions.

In this paper, we consider Cheeger's constant and the first eigenvalue of the nonlinear Laplacian on closed Finsler manifolds. Being based on these, we establish Cheeger's inequality and Buser's inequality for closed Finsler manifolds.

Estimates eigenvalues and spectrum for graph substructures using isocapacitary constants.

A new flow connects manifold invariants with critical exponents.

Study convex functions on manifolds without focal points, deriving new spectral properties.

New metric measure space theory for Lipschitz constants.

The paper bounds Cheeger ratios of eigenfunctions and their level sets.

We introduce notions of Cheeger constants for graphons and graphings. We prove Cheeger and Buser inequalities for these. On the way we prove co-area formulae for graphons and graphings.

Paper bounds the A-hat genus using curvature and isoperimetric constants.

In this paper, we study eigenvalues and eigenfunctions of $p$-Laplacians with Dirichlet boundary condition on graphs. We characterize the first eigenfunction (and the maximum eigenfunction for a bipartite graph) via the sign condition. By the uniqueness of the first eigenfunction of $p$-Laplacian, as $p\to 1,$ we ident…

We compute the Cheeger constants of a collection of hyperbolic surfaces corresponding to maximal non-compact arithmetic Fuchsian groups, and to subgroups which are the rotation subgroup of maximal reflection groups. The Cheeger constants are geometric quantities, but relate to the smallest eigenvalues of Maass cusp for…

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…

The paper studies eigenvalue problems on manifolds and recovers known inequalities.