相关论文: Infinite volume and atoms at the bottom of the spe…

For a closed minimal submanifold $f:M^n\looparrowright \mathbb{S}^{N}$ in the unit sphere $(n<N)$, we prove $${\rm Vol}(M^n) \geq\frac{n+1}{n+2}\int_{M}\left( 1+\varphi_{p}^2\right) \geq m{\rm Vol}(\mathbb{S}^{n}),$$ where…

This article studies the volume of compact quotients of reductive homogeneous spaces. Let $G/H$ be a reductive homogeneous space and $\Gamma$ a discrete subgroup of $G$ acting properly discontinuously and cocompactly on $G/H$. We prove that…

Let $ G $ be a real simple linear connected Lie group of real rank one. Then, $ X := G/K $ is a Riemannian symmetric space with strictly negative sectional curvature. By the classification of these spaces, $X$ is a real/complex/quaternionic…

We prove the following entropy-rigidity result in finite volume: if $X$ is a negatively curved manifold with curvature $-b^2\leq K_X \leq -1$, then $Ent_{top}(X) = n-1$ if and only if $X$ is hyperbolic. In particular, if $X$ has the same…

Let $X$ be a finite-area non-compact hyperbolic surface. We study the spectrum of the Laplacian on random covering surfaces of X and on random unitary bundles over X. We show that there is a constant $c > 0$ such that, with probability…

We study the spectrum of the Laplace operator of a complete minimal properly immersed hypersurface $M$ in $\R^{n+1}$. (1) Under a volume growth condition on extrinsic balls and a condition on the unit normal at infinity, we prove that $M$…

Let $G$ be a noncompact semisimple Lie group, $\Gamma$ be an irreducible cocompact lattice in $G$, and $P<G$ be a minimal parabolic subgroup. We consider the dynamics of $P$ acting on $G/\Gamma$ by left translation. For any infinite subset…

Let $\Gamma < G := \operatorname{SO}(d+1, 1)$ for $d \geq 1$ be a Zariski dense, geometrically finite, discrete subgroup with critical exponent strictly greater than $d/2$. We show that $L^2(\Gamma\backslash G)$ admits a strong spectral…

Let $G=\operatorname{SO}^\circ(n,1) \times \operatorname{SO}^\circ(n,1)$ and $X={\mathbb H}^{n}\times {\mathbb H}^{n}$ for $n\ge 2$. For a pair $(\pi_1, \pi_2)$ of non-elementary convex cocompact representations of a finitely generated…

In this paper we derive an explicit formula for the Hirzebruch-Mumford volume of an indefinite lattice L of rank at least 3. If \Gamma is an arithmetic subgroup of the group O(L) of isometries of L and L has signature (2,n), then an…

We consider operators arising from regular Dirichlet forms with vanishing killing term. We give bounds for the bottom of the (essential) spectrum in terms of exponential volume growth with respect to an intrinsic metric. As special cases we…

We show that discrete stationary random subgroups of isometry groups of Gromov hyperbolic spaces have full limit sets as well as critical exponents bounded from below. This information is used to answer a question of Gelander and show that…

Suppose $N$ is a diffuse, property T von Neumann algebra and X is an arbitrary finite generating set of selfadjoint elements for N. By using rigidity/deformation arguments applied to representations of N in full matrix algebras, we deduce…

We extend the notion of the cardinality of a discrete groupoid (equal to the Euler characteristic of the corresponding discrete orbifold) to the setting of Lie groupoids. Since this quantity is an invariant under equivalence of groupoids,…

We conjecture the following entropy bound to be valid in all space-times admitted by Einstein's equation: Let A be the area of any two-dimensional surface. Let L be a hypersurface generated by surface-orthogonal null geodesics with…

We investigate properties of the pseudo-Riemannian volume, entropy, and diameter for convex cocompact representations $\rho : \Gamma \to \mathrm{SO}(p,q+1)$ of closed $p$-manifold groups. In particular: We provide a uniform lower bound of…

Let $(X,g)$ be a complete noncompact geometrically finite surface with pinched negative curvature $-b^2\leq K_g \leq -1$. Let $\lambda_0(\widetilde{X})$ denote the bottom of the $L^2-$spectrum of the Laplacian on the universal cover…

Let $G$ be a free-by-cyclic group or a 2-dimensional right-angled Artin group. We provide an algebraic and a geometric characterization for when each aspherical simplicial complex with fundamental group isomorphic to $G$ has minimal volume…

It is classically known that closed geodesics on a compact Riemann surface with a metric of negative curvature strictly minimize length in their free homotopy class. We'd like to generalize this to Lagrangian submanifolds in K\"ahler…

Let X be a manifold equipped with a complete Riemannian metric of constant negative curvature and finite volume. We demonstrate the finiteness of the collection of totally geodesic immersed hypersurfaces in X that lie in the zero-level set…