Related papers: Simple geodesics on a punctured surface

Discrete forms of the mean and directed curvature are constructed on piecewise flat manifolds, providing local curvature approximations for smooth manifolds embedded in both Euclidean and non-Euclidean spaces. The resulting expressions take…

We examine closed geodesics in the quotient of hyperbolic three space by the discrete group of isometries SL(2,Z[i]). There is a correspondence between closed geodesics in the manifold, the complex continued fractions originally studied by…

In this paper, we obtain upper bounds on the multiplicity of Laplacian eigenvalues for closed hyperbolic surfaces in terms of the number of short closed geodesics and the genus $g$. For example, we show that if the number of short closed…

We modify the deformation method explored previously in a joint work of B. Shiffman and the author, in order to construct further examples of Kobayashi hyperbolic surfaces in the projective 3-space of any even degree starting with degree 8.

We consider higher-dimensional generalizations of the $\alpha$-Grushin plane, focusing on the problem of classification of geodesics that minimize length, also known as optimal synthesis. Solving Hamilton's equations on these spaces using…

We study deformations of complex hyperbolic surfaces which furnish the simplest examples of: (i) negatively curved K\"ahler manifolds and (ii) negatively curved Riemannian manifolds not having {\it constant} curvature. Although such complex…

This paper determines which orientable hyperbolic 3-manifolds contain simple closed geodesics. The Fuchsian group corresponding to the thrice-punctured sphere generates the only example of a complete non-elementary orientable hyperbolic…

It is proved that a bijection between two compact hyperbolic surfaces with boundary is an isometry if it and its inverse map each geodesic onto some geodesic.

Deciding realizability of a given polyhedral map on a (compact, connected) surface belongs to the hard problems in discrete geometry, from the theoretical, the algorithmic, and the practical point of view. In this paper, we present a…

For the minimal graph with strict convex level sets, we find an auxiliary function to study the Gaussian curvature of the level sets. We prove that this curvature function is a concave function with respect to the height of the minimal…

By recasting metrical geometry in a purely algebraic setting, both Euclidean and non-Euclidean geometries can be studied over a general field with an arbitrary quadratic form. Both an affine and a projective version of this new theory are…

We study invariant surfaces generated by one-parameter subgroups of simply and pseudo isotropic rigid motions. Basically, the simply and pseudo isotropic geometries are the study of a three-dimensional space equipped with a rank 2 metric of…

We exhibit orbits of the geodesic flow on a hyperbolic surface with at least one cusp such that every tubular neighborhood contains uncountably many distinct geodesic flow orbits. The proof relies on new phenomena, namely the existence of…

A numerical scheme is developed for solution of the Goursat problem for a class of nonlinear hyperbolic systems with an arbitrary number of independent variables. Convergence results are proved for this difference scheme. These results are…

We consider real isotropic geodesics on manifolds endowed with a pseudoconformal structure and their applications to the theory of lightlike hypersurfaces on such manifolds, the geometry of four-dimensional conformal structures of…

We introduce a novel representation and optimization framework for discrete geodesics on triangle meshes that reduces artifacts of linear methods on uneven and coarse discretizations. Our method computes squared geodesic distances from…

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…

Line fields on surfaces are a means to describe the nematic order that may pattern them. The least distorted nematic fields are called uniform, but they can only exist on surfaces with negative constant Gaussian curvature. To identify the…

We prove Fourier restriction estimates by means of the polynomial partitioning method for compact subsets of any sufficiently smooth hyperbolic hypersurface in threedimensional euclidean space. Our approach exploits in a crucial way the…

Cooper and Long generalised Epstein and Penner's Euclidean cell decomposition of cusped hyperbolic manifolds of finite volume to non-compact strictly convex projective manifolds of finite volume. We show that Weeks' algorithm to compute…