Related papers: Quantum Ergodicity for Eisenstein functions

Using quadratic forms, we stablish a criteria to relate the curvature of a Riemannian manifold and partial hyperbolicity of its geodesic flow. We show some examples which satisfy the criteria and another which does not satisfy it but still…

We investigate the existence, convergence and uniqueness of modified general curvature flow of convex hypersurfaces in hyperbolic space with a prescribed asymptotic boundary.

Let $S_{g,n}$ be a surface of genus $g > 1$ with $n>0$ punctures equipped with a complete hyperbolic cusp metric. Then it can be uniquely realized as the boundary metric of an ideal Fuchsian polyhedron. In the present paper we give a new…

We prove a new quantum variance estimate for toral eigenfunctions. As an application, we show that, given any orthonormal basis of toral eigenfunctions and any smooth embedded hypersurface with nonvanishing principal curvatures, there…

Continuing the work in \cite{ergodic-infinite}, we show that within each stratum of translation surfaces, there is a residual set of surfaces for which the geodesic flow in almost every direction is ergodic for almost-every periodic group…

Using systematically isothermal coordinates we show that there exist three different maximal extensions of the original Einstein-Rosen bridge. One of them, the hyperbolic Einstein-Rosen bridge, has two-dimensional sections diffeomorphic to…

We prove the existence of complete, embedded, constant mean curvature 1 surfaces in 3 dimensional hyperbolic space when g, the genus of the surface, and n, the number of ends of the surface, satisfy either g=0 and $n\geq 1$ or $g \geq 1$…

We construct new examples of Einstein metrics by perturbing the conformal infinity of geometrically finite hyperbolic metrics and by applying the inverse function theorem in suitable weighted H\"older spaces.

We study singularities of surfaces which are given by Kenmotsu-type formula with prescribed unbounded mean curvature.

For the quermassintegral inequalities of horospherically convex hypersurfaces in the $(n+1)$-dimensional hyperbolic space, where $n\geq 2$, we prove a stability estimate relating the Hausdorff distance to a geodesic sphere by the deficit in…

We obtain a gradient estimate for the Gauss maps from complete spacelike constant mean curvature hypersurfaces in Minkowski space into the hyperbolic space. As applications, we prove a Bernstein theorem which says that if the image of the…

Spacelike surfaces in Generalized Robertson-Walker spacetimes whose mean curvature function satisfies a natural nonlinear inequality are analyzed. Several uniqueness and nonexistence results for such compact spacelike surfaces are proved.…

In this paper, new sharp bounds for circular functions are proved. We provide some improvements of previous results by using infinite products, power series expansions and a generalisation of the so-called Bernoulli inequality. New proofs,…

In this series, we investigate quantum ergodicity at small scales for linear hyperbolic maps of the torus ("cat maps'"). In Part II of the series, we construct quasimodes that are quantum ergodic but are not equidistributed at the…

This is a tale describing the large scale geometry of Euclidean plane domains with their hyperbolic or quasihyperbolic distances. We prove that in any hyperbolic plane domain, hyperbolic and quasihyperbolic quasi-geodesics are the same…

We prove the existence of a hyperbolic surface spread over the sphere for which the projection map has all its singular values on the extended real line, and such that the preimage of the extended real line under the projection map is…

In this paper we continue our study of finding the curvature flow of complete hypersurfaces in hyperbolic space with a prescribed asymptotic boundary at infinity. Our main results are proved by deriving a priori global gradient estimates…

We show that a Kleinian surface group, or hyperbolic 3-manifold with a cusp-preserving homotopy-equivalence to a surface, has bounded geometry if and only if there is an upper bound on an associated collection of coefficients that depend…

We investigate some characteristic properties of specific Weingarten surfaces in the three-dimensional Euclidean space using the nets of the lines of curvature resp. the asymptotic lines on both central surfaces of them.

In this paper we extend Thurston's hyperbolic Dehn surgery theorem to a class of geometrically infinite hyperbolic 3-manifolds. As an application we prove a modest density theorem for Kleinian groups. We also discuss hyperbolic Dehn surgery…