Related papers: Short Simple Orthogonal Geodesic Chords on a 2-Dis…

We show that the geodesic period spectrum of a Riemannian 2-orbifold all of whose geodesics are closed depends, up to a constant, only on its orbifold topology and compute it. In the manifold case we recover the fact proved by Gromoll,…

The simple length spectrum of a Riemannian manifold is the set of lengths of its simple closed geodesics. We prove a theorem claimed by Lusternik: in any Riemannian 2-sphere whose simple length spectrum consists of only one element L, any…

Any finite configuration of curves with minimal intersections on a surface is a configuration of shortest geodesics for some Riemannian metric on the surface. The metric can be chosen to make the lengths of these geodesics equal to the…

In many singular metric spaces, the regularity of a shortest-length curve is unknown. Algebraic varieties, or more generally sets defined by finitely many polynomial or real analytic equalities or inequalities, all locally partition into…

We consider the problem of covering the boundary of a simple polygon on n vertices using the minimum number of geodesic unit disks. We present an O(n \log^2 n+k) time 2-approximation algorithm for finding the centers of the disks, with k…

Suppose that $M$ is a $2$-dimensional oriented Riemannian manifold, and let $\gamma$ be a simple closed curve on $M$. Let $m \gamma$ denote the curve formed by tracing $\gamma$ $m$ times. We prove that if $m \gamma$ is contractible through…

We study "flat knot types" of geodesics on compact surfaces M^2. For every flat knot type and any Riemannian metric g we introduce a Conley index associated with the curve shortening flow on the space of immersed curves on M^2. We conclude…

We use PDE methods as developed for the Liouville equation to study the existence of conformal metrics with prescribed singularities on surfaces with boundary, the boundary condition being constant geodesic curvature. Our first result shows…

The study of solutions with fixed energy of certain classes of Lagrangian (or Hamiltonian) systems is reduced, via the classical Maupertuis--Jacobi variational principle, to the study of geodesics in Riemannian manifolds. We are interested…

We show that the shortest closed geodesic on a 2-sphere with non-negative curvature has length bounded above by three times the diameter. We prove a new isoperimetric inequality for 2-spheres with pinched curvature; this allows us to…

We prove the existence of a constant $C > 0$ such that for any Riemannian metric $g$ on a 2-dimensional sphere $S^2$, there exist two distinct closed geodesics with lengths $L_{1}$ and $L_{2}$ satisfying $L_{1} L_{2} \leq C \cdot…

We prove that the unique least-perimeter way of partitioning the unit 2-dimensional disk into three regions of prescribed areas is by means of the standard graph consisting in three balanced constant geodesic curvature curves meeting…

Using an estimate on the number of critical points for a Morse-even function on the sphere $\mathbb S^m$, $m\ge1$, we prove a multiplicity result for orthogonal geodesic chords in Riemannian manifolds with boundary that are diffeomorphic to…

We prove several results concerning the existence of surfaces of section for the geodesic flows of closed orientable Riemannian surfaces. The surfaces of section $\Sigma$ that we construct are either Birkhoff sections, meaning that they…

We prove that, if a closed geodesic $\Gamma$ on a complete finite type hyperbolic surface has at least 2 self-intersections, then the length of $\Gamma$ has an lower bound $2\log(5+2\sqrt6)$, and the lower bound is sharp, attained on a…

We provide an easy approach to the geodesic distance on the general linear group GL(n) for left-invariant Riemannian metrics which are also right-O(n)-invariant. The parametrization of geodesic curves and the global existence of length…

We consider the following questions: given a hyperbolic plane domain and a separation of its complement into two disjoint closed sets each of which contains at least two points, what is the shortest closed hyperbolic geodesic which…

We show that, on an oriented compact surface, two sufficiently $C^2$-close Riemannian metrics with strictly convex boundary, no conjugate points, hyperbolic trapped set for their geodesic flows, and same marked boundary distance, are…

Suppose $M$ is a complete, non-compact $n$-dimensional Riemannian manifold with locally convex ends and finite volume. We prove that $M$ admits a non-trivial geodesic net with one vertex, at most $(n+2)(n+1)/2$ edges, and total length at…

We prove that for any two closed Riemannian manifolds $M^{2m}$ ($m\geq 1$) and $N$, there exists a minimizing (extrinsic) $m$-polyharmonic map for every free homotopy class in $[M^{2m}, N]$, provided that the homotopy group $\pi_{2m}(N)$ is…