Related papers: Prising apart geodesics by length in hyperbolic 3-…

We prove that a curve of degree $dk$ on a very general surface of degree $d \geq 5$ in $\mathbb{P}^3$ has geometric genus at least $\frac{dk(d-5)+k}{2} + 1$. This improves bounds given by G. Xu. As a corollary, we conclude that the very…

Let $S$ be a minimal surface of general type with irregularity $q(S) = 1$. Well-known inequalities between characteristic numbers imply that $3 p_g(S) \le c_2(S) \le 10 p_g(S)$, where $p_g(S)$ is the geometric genus and $c_2(S)$ the…

In this note we develop a tool box of non-Euclidean plane geometry methods that yield a constructive way to define in terms of closed geodesics the Goldman bracket on deformation classes of closed, directed curves. We use this construction…

In this paper we present a necessary conditions, that simple close geodesics on regular tetrahedra in the 3-dimensional hyperbolic space must satisfy. Furthermore, we explicitly describe three classes of simple closed geodesics on regular…

We consider hyperbolic projections of orbits of holomorphic self-maps of the unit disc, onto curves landing on the unit circle with a given angle. We show that under certain, necessary, assumptions, the projections exhibit monotonicity…

This article deals with the set of closed geodesics on complete finite type hyperbolic surfaces. For any non-negative integer $k$, we consider the set of closed geodesics that self-intersect at least $k$ times, and investigate those of…

We show that cusped finite-volume hyperbolic 3-manifolds contain infinitely many simple closed geodesics.

Let $M$ be a compact oriented 3-manifold with non-empty boundary consisting of surfaces of genii $>1$ such that the interior of $M$ is hyperbolizable. We show that for each spherical cone-metric $d$ on $\partial M$ such that all cone-angles…

We construct compact hyperbolic 3-manifolds with totally geodesic boundary, such that the closed 3-pseudomanifolds obtained by coning off the boundary components are negatively curved and contain locally convex subspaces whose fundamental…

We consider foliations of the whole three dimensional hyperbolic space H^3 by oriented geodesics. Let L be the space of all the oriented geodesics of H^3, which is a four dimensional manifold carrying two canonical pseudo-Riemannian metrics…

We investigate the geometry of $\pi_1$-injective surfaces in closed hyperbolic 3-manifolds. First we prove that for any $e>0$, if the manifold $M$ has sufficiently large systole $\sys_1(M)$, the genus of any such surface in $M$ is bounded…

We describe several methods to construct minimal foliations by hyperbolic surfaces on closed 3-manifolds, and discuss the properties of the examples thus obtained.

Submanifolds in Lorentz-Minkowski space are investigated from various mathematical viewpoints and are of interest also in relativity theory. We define the hyperbolic surface and the de Sitter surface of a curve in the spacelike hypersurface…

Let M be a compact, orientable, mean convex 3-manifold with boundary. We show that the set of all simple closed curves in the boundary of M which bound unique area minimizing disks in M is dense in the space of simple closed curves in the…

We define winding numbers of regular closed curves on surfaces with a nice euclidean or hyperbolic geometry. We prove that two regular closed curves are regularly homotopic if and only if they are freely homotopic and have the same winding…

Let M be an orientable hyperbolic surface without boundary and let $\gamma$ be a closed geodesic in M. We prove that any side of any triangle formed by distinct lifts of $\gamma$ in H2 is shorter than $\gamma$.

We investigate the geometry of closed, orientable, hyperbolic $3$-manifolds whose fundamental groups are $k$-free for a given integer $k\ge 3$. We show that any such manifold $M$ contains a point $P$ of $M$ with the following property: If…

We study conservative partially hyperbolic diffeomorphisms in hyperbolic 3-manifolds. We show that they are always accessible and deduce as a result that every conservative $C^{1+}$ partially hyperbolic in a hyperbolic 3-manifold must be…

We consider closed hypersurfaces smoothly immersed in hyperbolic manifolds up to homotopy and commensurability. We prove that if a closed hyperbolic manifold $M$ contains a sequence of asymptotically geodesic hypersurfaces, then $\pi_1(M)$…

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…