Related papers: Large Steklov eigenvalues on hyperbolic surfaces

In this paper, we study the higher Steklov eigenvalues of graphs on surfaces. We obtain the upper bound of higher Steklov eigenvalues of a finite graph $G$ with boundary $B$ and genus $g$ by using metrical deformation via probability flows.…

In recent years, eigenvalue optimization problems have received a lot of attention, in particular, due to their connection with the theory of minimal surfaces. In the present paper we prove that on any orientable surface there exists a…

We construct surfaces with arbitrarily large multiplicity for their first non-zero Steklov eigenvalue. The proof is based on a technique by M. Burger and B. Colbois originally used to prove a similar result for the Laplacian spectrum. We…

In this paper we obtain several results concerning the optimization of higher Steklov eigenvalues both in two and higher dimensional cases. We first show that the normalized (by boundary length) $k$-th Steklov eigenvalue on the disk is not…

We study geometric and spectral properties of typical hyperbolic surfaces of high genus, excluding a set of small measure for the Weil-Petersson probability measure. We first prove Benjamini-Schramm convergence to the hyperbolic plane H as…

Let $(M,g,\sigma)$ be a compact Riemmannian surface equipped with a spin structure $\sigma$. For any metric $\tilde{g}$ on $M$, we denote by $\mu\_1(\tilde{g})$ (resp. $\lambda\_1(\tilde{g})$) the first positive eigenvalue of the Laplacian…

We prove two explicit bounds for the multiplicities of Steklov eigenvalues $\sigma_k$ on compact surfaces with boundary. One of the bounds depends only on the genus of a surface and the index $k$ of an eigenvalue, while the other depends as…

Let $G$ be a graph of genus $g$ with boundary $\delta\Omega$. For $g=0$, Lin and Zhao [J. Lond. Math. Soc. 112 (2025), Paper No. e70238] proved an upper bound for the first (non-trivial) Steklov eigenvalue of $(G, \delta\Omega )$, and they…

This article introduces the notion of L-tangle-free compact hyperbolic surfaces, inspired by the identically named property for regular graphs. Random surfaces of genus g, picked with the Weil-Petersson probability measure, are (a log…

We prove existence and regularity of metrics on a surface with boundary which maximize sigma_1 L where sigma_1 is the first nonzero Steklov eigenvalue and L the boundary length. We show that such metrics arise as the induced metrics on free…

We give explicit isoperimetric upper bounds for all Steklov eigenvalues of a compact orientable surface with boundary, in terms of the genus, the length of the boundary, and the number of boundary components. Our estimates generalize a…

In this article we study the asymptotic behavior of small eigenvalues of Riemann surfaces for large genus. We show that for any positive integer $k$, as the genus $g$ goes to infinity, the smallest $k$-th eigenvalue of Riemann surfaces in…

Let $(\Omega^{n+1}, g)$ be an $(n+1)$-dimensional smooth compact connected Riemannian manifold with smooth boundary $\Sigma$, satisfying that ${\text{Ric}_{\Omega}}\ge 0$ and $\Sigma$ is strictly convex, more precisely, its second…

Let $\Delta$ and $B$ be the maximum vertex degree and a subset of vertices in a graph $G$ respectively. In this paper, we study the first (non-trivial) Steklov eigenvalue $\sigma_2$ of $G$ with boundary $B$. Using metrical deformation via…

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 study the geometry and spectral theory of Weil-Petersson random surfaces with genus-$g$ and $n$ cusps in the large-$n$ limit. We show that for a random hyperbolic surface in $\mathcal{M}_{g,n}$ with $n$ large, the number of small…

In this article we introduce a hyperbolic metric on the (normalized) space of stability conditions on projective K3 surfaces $X$ with Picard rank $\rho (X) =1$. And we show that all walls are geodesic in the normalized space with respect to…

Let $\mathcal{C}(S_{g,p})$ denote the curve complex of the closed orientable surface of genus $g$ with $p$ punctures. Masur-Minksy and subsequently Bowditch showed that $\mathcal{C}(S_{g,p})$ is $\delta$-hyperbolic for some…

We give results on optimal constants of isoperimetric inequalities involving Steklov eigenvalues on surfaces with boundary. We both consider this question on Riemannian surfaces with a same given topology or more specifically belonging to…

We prove (Theorem~1.5) that there exists a constant $\Lambda > 0$ so that if $M$ is a $(\mu,d)$-generic complete hyperbolic 3-manifold of volume $\vol[M] < \infty$ and $\Sigma \subset M$ is a Heegaard surface of genus $g(\Sigma) > \Lambda…