Related papers: The intersection dual of geodesic currents

The consideration of the so-called rotation minimizing frames allows for a simple and elegant characterization of plane and spherical curves in Euclidean space via a linear equation relating the coefficients that dictate the frame motion.…

The geodesics in the group of volume-preserving diffeomorphisms (volumorphisms) of a manifold $M$, for a Riemannian metric defined by the kinetic energy, can be used to model the movement of ideal fluids in that manifold. The existence of…

In this paper we study the ergodic theory and thermodynamic formalism of the geodesic flow on non-compact pinched negatively curved manifolds. We consider two notions of entropy at infinity, the topological and the measure theoretic entropy…

Recently the authors showed that there is a robust potential theory attached to any calibrated manifold (X,\phi). In particular, on X there exist \phi-plurisubharmonic functions, \phi-convex domains, \phi-convex boundaries, etc., all…

Motivated by Gromov's geodesic flow problem on hyperbolic groups $G$, we develop in this paper an analog using random walks. This leads to a notion of a harmonic analog $\Theta$ of the Bowen-Margulis-Sullivan measure on $\partial^2 G$. We…

We consider hyperbolic structures on the compression body C with genus 2 positive boundary and genus 1 negative boundary. Note that C deformation retracts to the union of the torus boundary and a single arc with its endpoints on the torus.…

We study configurations of immersed curves in surfaces and surfaces in 3-manifolds. Among other results, we show that primitive curves have only finitely many configurations which minimize the number of double points. We give examples of…

If X is a geodesic metric space and $x_1,x_2,x_3\in X$, a {\it geodesic triangle} $T=\{x_1,x_2,x_3\}$ is the union of the three geodesics $[x_1x_2]$, $[x_2x_3]$ and $[x_3x_1]$ in $X$. The space $X$ is $\delta$-\emph{hyperbolic} $($in the…

We prove the existence of Alexandrov embedded closed magnetic geodesics on closed hyperbolic surfaces. Closed magnetic geodesics correspond to closed curves with prescribed geodesic curvature.

Starting with a pseudo-Anosov flow $\varphi$ on a closed hyperbolic $3$-manifold $M$ and an embedded surface $S \subset M$ that is (almost) transverse to $\varphi$, we relate the hyperbolic geometry of $M$ (e.g. volume, circumference, short…

We introduce the functor * which assigns to every metric space X its symmetric join *X. As a set, *X is a union of intervals connecting ordered pairs of points in X. Topologically, *X is a natural quotient of the usual join of X with…

In this paper we study the ergodic theory of the geodesic flow on negatively curved geometrically finite manifolds. We prove that the measure theoretic entropy is upper semicontinuous when there is no loss of mass. In case we are losing…

We describe, in an intrinsic way and using the global chart provided by Ito's parallel transport, a generalisation of the notion of geodesic (as critical path of an energy functional) to diffusion processes on Riemannian manifolds. These…

We give a natural definition of geodesics on a Riemannian supermanifold and extend the usual geodesic flow defined on the cotangent bundle of the body of the supermanifold, associated to the induced Riemannian structure on the body, to a…

The author shows that equicontinuous geodesic flows on surfaces are periodic. A similar result for flows on 3-manifolds is also proven. The idea of the proof is to show that the return map is recurrent and therefore periodic.

Identifying parallel sides of a collection of Euclidean polygons yields a flat surface with cone points of angles multiples of 2 pi, naturally a compact Riemann surface but also an algebraic curve, and a hyperbolic surface. In general two…

Superconducting systems that simultaneously lack space-inversion and time-reversal symmetries have recently been the subject of a flurry of experimental and theoretical research activities. Their ability to carry supercurrents with…

The ultimate goal of our book is to present a unified approach to the dynamics, ergodic theory, and geometry of elliptic functions from $\C$ to $\oc$. We consider elliptic functions as a most regular class of transcendental meromorphic…

We study flows generated within a two-dimensional corner by the chemical activity of the confining boundaries. Catalytic reactions at the surfaces induce diffusioosmotic motion of the viscous fluid throughout the domain. The presence of…

In this article we obtain a simple topological and dynamical systems condition which is necessary and sufficient for an arbitrary pseudo-Anosov flow in a closed, hyperbolic three manifold to be quasigeodesic. Quasigeodesic means that orbits…