Related papers: Beyond almost Fuchsian space

We consider the space of all quasifuchsian metrics on the product of a surface with the real line. We show that, in a neighborhood of the submanifold consisting of fuchsian metrics, every non-fuchsian metric is completely determined by the…

Let F be a surface and suppose that \phi: F -> F is a pseudo-Anosov homeomorphism fixing a puncture p of F. The mapping torus M = M_\phi is hyperbolic and contains a maximal cusp C about the puncture p. We show that the area (and height) of…

Geometrization theorem, fibered case: Every three-manifold that fibers over the circle admits a geometric decomposition. Double limit theorem: for any sequence of quasi-Fuchsian groups whose controlling pair of conformal structures tends…

In this short note we prove that a Kahler manifold with lower Ricci curvature bound and almost maximal volume is Gromov-Hausdorff close to the projective space with the Fubini-Study metric. This is done by combining the recent results of…

In this paper we discuss when a quasi-conformally flat weakly Ricci symmetric manifold (of dimension greater than 3) becomes a manifold of hyper quasi-constant curvature, a quasi-Einstein manifold and a manifold of quasi-constant curvature.…

We show that in Cartan-Hadamard manifolds $M^n$, $n\geq 3$, closed infinitesimally convex hypersurfaces $\Gamma$ bound convex flat regions, if curvature of $M^n$ vanishes on tangent planes of $\Gamma$. This encompasses…

Let $M$ be a quasi-Fuchsian three-manifold that contains a closed incompressible surface with principal curvatures within the range of the unit interval, for a prescribed function $H$ (with mild conditions) on $M$, we construct a closed…

We consider properly discontinuous, isometric, convex cocompact actions of surface groups on a CAT(-1) space. We show that the limit set of such an action, equipped with the canonical visual metric, is a (weak) quasicircle in the sense of…

Almost toric manifolds form a class of singular Lagrangian fibered symplectic manifolds that is a natural generalization of toric manifolds. Notable examples include the K3 surface, the phase space of the spherical pendulum and rational…

We prove a version of the strong half-space theorem between the classes of recurrent minimal surfaces and complete minimal surfaces with bounded curvature of $\mathbb{R}^{3}_{\raisepunct{.}}$ We also show that any minimal hypersurface…

Surfaces of finite geometric type are complete, immersed into the tree-dimensional Euclidean space with finite total curvature and Gauss map extending to an oriented compact surface as a smooth branched covering map over the unit sphere of…

About a decade ago Thurston proved that a vast collection of 3-manifolds carry metrics of constant negative curvature. These manifolds are thus elements of {\em hyperbolic geometry}, as natural as Euclid's regular polyhedra. For a closed…

A sequence of distinct closed surfaces in a hyperbolic 3-manifold M is asymptotically geodesic if their principal curvatures tend uniformly to zero. When M has finite volume, we show such sequences are always asymptotically dense in the…

In this paper we study the rigidity of infinite volume 3-manifolds with sectional curvature $-b^2\le K\le -1$ and finitely generated fundamental group. In-particular, we generalize the Sullivan's quasi-conformal rigidity for finitely…

A three-dimensional quasi-Fuchsian Lorentzian manifold $M$ is a globally hyperbolic spacetime diffeomorphic to $\Sigma\times (-1,1)$ for a closed orientable surface $\Sigma$ of genus $\geq 2$. It is the quotient $M=\Gamma\backslash…

We consider complete noncompact Riemannian manifolds with quadratically decaying lower Ricci curvature bounds and minimal volume growth. We first prove a rigidity result showing that ends with strongly minimal volume growth are isometric to…

A compact Fuchsian manifold with boundary is a hyperbolic 3-manifold homeomorphic to $S_g \times [0; 1]$ such that the boundary component $S_g \times \{ 0\}$ is geodesic. We prove that a compact Fuchsian manifold with convex boundary is…

We study principal curvatures of fibers and Heegaard surfaces smoothly embedded in hyperbolic 3-manifolds. It is well known that a fiber or a Heegaard surface in a hyperbolic 3-manifold cannot have principal curvatures everywhere less than…

Let N be a manifold (with boundary) of dimension at least 3, such that its interior admits a hyperbolic metric of finite volume. We discuss the possible limits arising from sequences of relative fundamental cycles approximating the…

In this paper, we classify completely hyperbolic 3-manifolds corresponding to geometric limits of Kleinian surface groups isomorphic to $\pi_1(S)$ for a finite-type hyperbolic surface $S$. In the first of the three main theorems, we…