相关论文: Cheeger's constant and the first eigenvalue of a c…

We prove estimates for the sharp constants in fractional Poincar\'e-Sobolev inequalities associated to an open set, in terms of a nonlocal capacitary extension of its inradius. This work builds upon previous results obtained in the local…

In this paper, we obtain the Bossel-Daners inequality for the first eigenvalue of the p-Laplacian with Robin boundary conditions on complete Riemannian manifolds with lower Ricci curvature bounds. Furthermore, we demonstrate that the…

We prove the stability of the Gieseker point of an irreducible homogeneous bundle over a rational homogeneous space. As an application we get a sharp upper estimate for the first eigenvalue of the Laplacian of an arbitrary Kaehler metric on…

In this paper we study the behavior of the second eigenfunction of the anisotropic $p$-Laplace operator \[ - Q_{p}u:=-\textrm{div} \left(F^{p-1}(\nabla u)F_\xi (\nabla u)\right), \] as $p \to 1^+$, where $F$ is a suitable smooth norm of…

We prove the sharp inequality \[ J(\Omega) := \frac{\lambda_1(\Omega)}{h_1(\Omega)^2} < \frac{\pi^2}{4},\] where $\Omega$ is any planar, convex set, $\lambda_1(\Omega)$ is the first eigenvalue of the Laplacian under Dirichlet boundary…

In this article we study the stability problem for positive quaternion-K\"ahler manifolds. We give a description of infinitesimal Einstein deformations and destabilising directions in terms of Laplace eigenfunctions and a special class of…

We give some sharp lower bounds of the first eigenvalue for the Hodge Laplacian acting on differential forms on the boundary of a Riemannian manifold. We also give some sharp estimates for the first nonzero Steklov eigenvalue for…

We study the Cheeger isoperimetric constant as a functional in the space of convex co-compact hyperbolic 3-manifolds which are either quasi-Fuchsian or acylindrical. We prove that the global maximum is attained uniquely at the Fuchsian…

Let $ M^n$ be a closed immersed minimal hypersurface in the unit sphere $\mathbb{S}^{n+1}$. We establish a special isoperimetric inequality of $M^n$. As an application, if the scalar curvature of $ M^n$ is constant, then we get a uniform…

In this paper we prove a sharp inequality of Szeg\H{o}-Weinberger type for the first nontrivial eigenvalue of the $p-$Laplacian with Neumann boundary conditions. This applies to convex sets with given diameter. Some variants, extensions and…

In this article, we establish a geometric lower bound for the first positive eigenvalue $\lambda^{(1)}_{1}$ of the rough Laplacian acting on $1$-forms for closed $2n$-dimensional Riemannian manifolds with nonvanishing Euler characteristic.…

Given a smooth positive function $f$ defined on the unit circle satisfying a simple condition, we obtain a Poincar\'{e}-type inequality for an arbitrary function $u$ whose weighted average with respect to $f$ is zero. The proof uses…

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…

We study a Dirichlet-to-Neumann eigenvalue problem for differential forms on a compact Riemannian manifold with smooth boundary. This problem is a natural generalization of the classical Steklov problem on functions. We derive a number of…

Using the definition of a Finsler--Laplacian given by the first author, we show that two bi-Lipschitz Finsler metrics have a controlled spectrum. We deduce from that several generalizations of Riemannian results. In particular, we show that…

In this paper, some new forms of the Cheeger's inequalities are established for general (maybe unbounded) symmetric forms, the resulting estimates improve and extend the ones obtained by Lawler and Sokal (1988) for bounded jump processes.…

We study the limit of first eigenfunctions of (discrete) $p$-Laplacian on a finite subset of a graph with Dirichlet boundary condition, as $p\to 1.$ We prove that up to a subsequence, they converge to a summation of characteristic functions…

We consider the magnetic Steklov eigenvalue problem on compact Riemannian manifolds with boundary for generic magnetic potentials and establish various results concerning the spectrum. We provide equivalent characterizations of magnetic…

We prove various estimates for the first eigenvalue of the magnetic Dirichlet Laplacian on a bounded domain in two dimensions. When the magnetic field is constant, we give lower and upper bounds in terms of geometric quantities of the…

In this paper, we consider a variation on Cheeger numbers related to the coboundary expanders recently defined by Dotterer and Kahle. A Cheeger-type inequality is proved, which is similar to a result on graphs due to Fan Chung. This…