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Our main result is that for all sufficiently large $x_0>0$, the set of commensurability classes of arithmetic hyperbolic 2- or 3-orbifolds with fixed invariant trace field $k$ and systole bounded below by $x_0$ has density one within the…

The main theorem of this article provides sufficient conditions for a degree $d$ finite cover $M'$ of a hyperbolic 3-manifold $M$ to be a surface-bundle. Let $F$ be an embedded, closed and orientable surface of genus $g$, close to a minimal…

In this note we prove three rigidity results for Einstein manifolds with bounded covering geometry. (1) An almost flat manifold $(M,g)$ must be flat if it is Einstein, i.e. $\operatorname{Ric}_g=\lambda g$ for some real number $\lambda$.…

We investigate shrinking maps from a cusped hyperbolic surface into the moduli space of closed Riemann surfaces. For such a map and its lift to the Teichm\"uller space, we consider whether they are quasi-isometric embeddings with respect to…

Let $M$ be a closed hyperbolic $3$-manifold. A homotopy class $[S]$ of surfaces in $M$ is filling if any representative cuts $M$ into components contractible in $M$. We prove that there exist $\epsilon_0, g_0>0$ such that every homotopy…

Abelian covers of hyperbolic $3$-manifolds are ubiquitous. We prove the local mixing theorem of the frame flow for abelian covers of closed hyperbolic $3$-manifolds. We obtain a classification theorem for measures invariant under the…

For a single cusped hyperbolic 3-manifold, Hodgson proved that there are only finitely many Dehn fillings of it whose trace fields have bounded degree. In this paper, we conjecture the same for manifolds with more cusps, and give the first…

Colding and Gabai have given an effective version of Li's theorem that non-Haken hyperbolic 3-manifolds have finitely many irreducible Heegaard splittings. As a corollary of their work, we show that Haken hyperbolic 3-manifolds have a…

Let M be a compact hyperbolic manifold with totally geodesic boundary. If the injectivity radius of the boundary is larger than an explicit function of the normal injectivity radius of the boundary, we show that there is a negatively curved…

We show that any star-shaped convex hypersurface with constant Weingarten curvature in the deSitter-Schwarzschild manifold is a sphere of symmetry. Moreover, we study an isoperimetric problem for bounded domains in the doubled Schwarzschild…

We prove the filling area conjecture in the hyperelliptic case. In particular, we establish the conjecture for all genus 1 fillings of the circle, extending P. Pu's result in genus 0. We translate the problem into a question about closed…

Let M be a closed hyperbolic three manifold. We construct closed surfaces which map by immersions into M so that for each one the corresponding mapping on the universal covering spaces is an embedding, or, in other words, the corresponding…

We show that any noncompact oriented surface is homeomorphic to the leaf of a minimal foliation of a closed $3$-manifold. These foliations are (or are covered by) suspensions of continuous minimal actions of surface groups on the circle.…

It is given a topological pinching for the injectivity radius of a compact embedded surface either in the sphere or in the hyperbolic space

Let $\mathcal{X}_S$ denote the class of spaces homeomorphic to two closed orientable surfaces of genus greater than one identified to each other along an essential simple closed curve in each surface. Let $\mathcal{C}_S$ denote the set of…

We show that a hyperbolic three-manifold $M$ containing a closed minimal surface with principal curvatures in $[-1,1]$ also contains nearby (non-minimal) surfaces with principal curvatures in $(-1,1)$. When $M$ is complete and homeomorphic…

In this paper we shall study smooth submanifolds immersed in a k-step Carnot group G of homogeneous dimension Q. Among other results, we shall prove an isoperimetric inequality for the case of a $C^2$-smooth compact hypersurface S with - or…

We prove that a stable minimal hypersurface of an open ball having a singular set of locally finite codimension 2 Hausdorff measure which is weakly close to a multiplicity 2 hyperplane is a 2-valued C^{1, alpha} graph in the interior.…

We study random covers of a closed hyperbolic surface $\Sigma$, subject to the condition that, for $k\geq 2$, the fundamental group is isomorphic to the free group $F_k$. We show that asymptotically they distribute according to a specific…

We prove the strong Weinstein conjecture for closed contact manifolds that appear as the concave boundary of a symplectic cobordism admitting an essential local foliation by holomorphic spheres.