Related papers: On Cheeger constants of hyperbolic surfaces

In this paper, we investigate the asymptotics of shortest filling closed multi-geodesics of closed hyperbolic surfaces as systole $\to 0$ or as genus $\to \infty$. We first show that for a closed hyperbolic surface $X_g$ of genus $g$, the…

A polygonal surface in the pseudo-hyperbolic space H^(2,n) is a complete maximal surface bounded by a lightlike polygon in the Einstein universe Ein^(1,n) with finitely many vertices. In this article, we give several characterizations of…

In this paper, we introduce Cheeger type constants via isocapacitary constants introduced by Maz'ya to estimate first Dirichlet, Neumann and Steklov eigenvalues on a finite subgraph of a graph. Moreover, we estimate the bottom of the…

The classical Cheeger's inequality relates the edge conductance $\phi$ of a graph and the second smallest eigenvalue $\lambda_2$ of the Laplacian matrix. Recently, Olesker-Taylor and Zanetti discovered a Cheeger-type inequality $\psi^2 /…

We relate the nontrivial singular values $\sigma_2,\ldots,\sigma_n$ of the normalized adjacency matrix of an Eulerian directed graph to combinatorial measures of graph expansion: \\ 1. We introduce a new directed analogue of conductance…

Extending Blaschke and Lebesgue's classical result in the Euclidean plane, it has been recently proved in spherical and the hyperbolic cases, as well, that Reuleaux triangles have the minimal area among convex domains of constant width $D$.…

The so-called {\it kissing number} for hyperbolic surfaces is the maximum number of homotopically distinct systoles a surface of given genus $g$ can have. These numbers, first studied (and named) by Schmutz Schaller by analogy with lattice…

We construct an exact expression for the site percolation threshold p_c on a quasi-regular tree, and a related exact lower bound for a quasi-regular graph. Both are given by the inverse spectral radius of the appropriate Hashimoto matrix…

We discuss an alternative approach to the uniformisation problem on surfaces with boundary by representing conformal structures on surfaces $M$ of general type by hyperbolic metrics with boundary curves of constant positive geodesic…

We complement our work on the causality of upper semi-continuous distributions of cones with some results on Cauchy hypersurfaces. We prove that every locally stably acausal Cauchy hypersurface is stable. Then we prove that the signed…

For a fixed natural number $t\geq 2$, we consider $t$-uniform random hypergraphs $\mathscr{H} (n,t,p)$ on $n$ vertices $[n]=\{1,\ldots, n\}$, where each $t$-subset of $[n]$ is included as a hyperedge with probability $p$ and independently.…

We perform a detailed analysis of the solvability of linear strain equations on hyperbolic surfaces to obtain $L^2$ regularity solutions. Then the rigidity results on the strain tensor of the middle surface are implied by the $L^2$…

This paper contains a purely topological theorem and a geometric application. The topological theorem states that if M is a simple closed orientable 3-manifold such that \pi_1(M) contains a genus g surface group and H_1(M;Z/2Z) has rank at…

Houdr\'e and Tetali defined a class of isoperimetric constants $\varphi_p$ of graphs for $0 \leq p \leq 1$, and conjectured a Cheeger-type inequality for $\varphi_\frac12$ of the form $$\lambda_2 \lesssim \varphi_\frac12 \lesssim…

The bondage number of a graph is the smallest number of its edges whose removal results in a graph having a larger domination number. We provide constant upper bounds for the bondage number of graphs on topological surfaces, improve upper…

We study $L^p$ boundedness of the maximal average over dilations of a smooth hypersurface $S$. When the decay rate of the Fourier transform of a measure on $S$ is $1/2$, we establish the optimal maximal bound, which settles the conjecture…

We discuss the minimization of a Kohler-Jobin type scale-invariant functional among open, convex, bounded sets, namely $\min T_2(\Omega) ^{\frac{1}{N+2}}h_1(\Omega)$ among open convex bounded sets $\Omega \subset \mathbb R^N$, where…

We introduce the quasi-hyperbolicity constant of a metric space, a rough isometry invariant that measures how a metric space deviates from being Gromov hyperbolic. This number, for unbounded spaces, lies in the closed interval $[1,2]$. The…

We consider the problem of finding the maximum number $e_d(n)$ of pairs of touching circles in a packing of $n$ congruent circles of diameter $d$ in the hyperbolic plane of curvature $-1$. In the Euclidean plane, the maximum comes from a…

The hypergraph regularity lemma -- the extension of Szemer\'edi's graph regularity lemma to the setting of $k$-uniform hypergraphs -- is one of the most celebrated combinatorial results obtained in the past decade. By now there are several…