arXiv research

New inequality for special forms on manifolds.

The paper proves inequalities for Steklov eigenvalues on finite graphs.

Solved Cheeger inequalities for simplicial complexes, combining topological and graph theoretic methods.

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.

Lower bounds for eigenvalues on manifolds with boundary conditions.

Study Cheeger inequalities for Riemannian manifolds with boundary.

Generalizes Cheeger inequality to Carnot-Carathéodory spaces.

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.

Sharp stability results for reverse isoperimetric inequalities in 2D.

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

Paper connects probability density cuts to graph theory eigenfunctions.

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…

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

Uniform waist inequalities proven for manifolds with Kazhdan groups in codimension two.

Sharp inequalities proved for RCD spaces, showing equality conditions.

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

Study shows convergence rates for Cheeger cuts on data clouds.

In this paper, we establish Buser type inequalities, i.e., upper bounds for eigenvalues in terms of Cheeger constants. We prove the Buser's inequality for an infinite but locally finite connected graph with Ricci curvature lower bounds. Furthermore, we derive that the graph with positive curvature is finite, especially…

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

The paper shows inequality and rigidity for manifolds with integral Ricci curvature.

Quantifies fractional isoperimetric inequality with strong control over boundary oscillation.

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

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…

Survey on rigidity and almost rigidity of Green functions in non-negative Ricci curvature spaces.

Let $(X,d,μ)$ be a complete metric measure space, with $μ$ a locally doubling measure, that supports a local weak $L^2$-Poincaré inequality. By assuming a heat semigroup type curvature condition, we prove that Cheeger-harmonic functions are Lipschitz continuous on $(X,d,μ)$. Gradient estimates for Cheeger-harmonic func…

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.

Using an inverse system of metric graphs as in: J. Cheeger and B. Kleiner, "Inverse limit spaces satisfying a Poincaré inequality", we provide a simple example of a metric space $X$ that admits Poincaré inequalities for a continuum of mutually singular measures.

We prove Cheeger inequalities for p-Laplacians on finite and infinite weighted graphs. Unlike in previous works, we do not impose boundedness of the vertex degree, nor do we restrict ourselves to the normalized Laplacian and, more generally, we do not impose any boundedness assumption on the geometry. This is achieved …

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…

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

Study on extremizers for Sobolev inequality on curved manifolds.

The paper studies isoperimetric inequalities on warped product manifolds.

Estimates eigenvalues and spectrum for graph substructures using isocapacitary constants.

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

Study finds eigenvalue bounds for non-convex domains using cohomology.

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 …

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

For a sequence $\{(M_i, g_i, x_i)\}$ of pointed Riemannian manifolds with boundary, the sequence $\{(M_i,\tilde g_i,x_i)\}$ is its conformal satellite if the metric $\tilde g_i$ is conformal to $g_i$, that is, $\tilde g_i=u^{\frac{4}{n-2}}_ig_i$. Assuming the manifolds $(M_i,g_i,x_i)$ have uniformly bounded geometry, w…

We study complexities of 3-manifolds defined from triangulations, Heegaard splittings, and surgery presentations. We show that these complexities are related by linear inequalities, by presenting explicit geometric constructions. We also show that our linear inequalities are asymptotically optimal. Our results are used…

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…

Paper tackles high-order inference in structured prediction tasks.

Let φ(G) be the minimum conductance of an undirected graph G, and let 0=λ_1 <= λ_2 <=... <= λ_n <= 2 be the eigenvalues of the normalized Laplacian matrix of G. We prove that for any graph G and any k >= 2, φ(G) = O(k) λ_2 / \sqrt{λ_k}, and this performance guarantee is achieved by the spectral partitioning algorithm. …

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…

ULES embeds dynamic networks with stability guarantees.

The study explores discrete versions of Riemannian geometry structures on manifolds.

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…

Buser's inequality gives an upper bound on the first non-zero eigenvalue of the Laplacian of a closed manifold M in terms of the Cheeger constant h(M). Agol later gave a quantitative improvement of Buser's inequality. Agol's result is less transparent since it is given implicitly by a set of equations, one of which is …

The paper establishes inequalities for minimal graphs on manifolds with nonnegative Ricci curvature.