Filling minimality of finslerian 2-discs

We prove that every Riemannian metric on the 2-disc such that all its geodesics are minimal is a minimal filling of its boundary (within the class of fillings homeomorphic to the disc). This improves an earlier result of the author by removing the assumption that the boundary is convex. More generally, we prove this result for Finsler metrics with area defined as the two-dimensional Holmes-Thompson volume. This implies a generalization of Pu’s isosystolic inequality to Finsler metrics, both for the Holmes-Thompson and Busemann definitions of the Finsler area.     

Sup norms of Cauchy data of eigenfunctions on manifolds with concave boundary

We prove that the Cauchy data of Dirichlet or Neumann Δ- eigenfunctions of Riemannian manifolds with concave (diffractive) boundary can only achieve maximal sup norm bounds if there exists a self-focal point on the boundary, i.e., a point at which a positive measure of geodesics leaving the point return to the point. In the case of real analytic Riemannian manifolds with real analytic boundary, maximal sup norm bounds on boundary traces of eigenfunctions can only be achieved if there exists a point on the boundary at which all geodesics loop back. As an application, the Dirichlet or Neumann eigenfunctions of Riemannian manifolds with concave boundary and non-positive curvature never have eigenfunctions whose boundary traces achieve maximal sup norm bounds. The key new ingredient is the Melrose–Taylor diffractive parametrix and Melrose’s analysis of the Weyl law.     

Spaces of domino tilings

We consider the set of all tilings by dominoes (2×1 rectangles) of a surface, possibly with boundary, consisting of unit squares. Convert this set into a graph by joining two tilings by an edge if they differ by aflip, i.e., a 90° rotation of a pair of side-by-side dominoes. We give a criterion to decide if two tilings are in the same connected component, a simple formula for distances, and a method to construct geodesics in this graph. For simply connected surfaces, the graph is connected. By naturally adjoining to this graph higher-dimensional cells, we obtain a CW-complex whose connected components are homotopically equivalent to points or circles. As a consequence, for any region different from a torus or Klein bottle, all geodesics with common endpoints are equivalent in the following sense. Build a graph whose vertices are these geodesics, adjacent if they differ only by the order of two flips on disjoint squares: this graph is connected. The first two authors received support from SCT and CNPq, Brazil. The other two were supported by a grant for undergraduates by CNPq.     

Multiple closed geodesics on 3-spheres

This paper is devoted to a study on closed geodesics on Finsler and Riemannian spheres. We call a prime closed geodesic on a Finsler manifold rational, if the basic normal form decomposition (cf. [Y. Long, Bott formula of the Maslov-type index theory, Pacific J. Math. 187 (1999) 113-149]) of its linearized Poincaré map contains no 2×2 rotation matrix with rotation angle which is an irrational multiple of π, or irrational otherwise. We prove that if there exists only one prime closed geodesic on a d-dimensional irreversible Finsler sphere with d?2, it cannot be rational. Then we further prove that there exist always at least two distinct prime closed geodesics on every irreversible Finsler 3-dimensional sphere. Our method yields also at least two geometrically distinct closed geodesics on every reversible Finsler as well as Riemannian 3-dimensional sphere. We prove also such results hold for all compact simply connected 3-dimensional manifolds with irreversible or reversible Finsler as well as Riemannian metrics.     

Connecting Geodesics and Security of Configurations in Compact Locally Symmetric Spaces

A pair of points in a Riemannian manifold makes a secure configuration if the totality of geodesics connecting them can be blocked by a finite set. The manifold is secure if every configuration is secure. We investigate the security of compact, locally symmetric spaces.     

Minimal rays on closed surfaces

Given an arbitrary Riemannian metric on a closed surface, we consider length-minimizing geodesics in the universal cover. Morse and Hedlund proved that such minimal geodesics lie in bounded distance of geodesics of a Riemannian metric of constant curvature. Knieper asked when two minimal geodesics in bounded distance of a single constant-curvature geodesic can intersect. In this paper we prove an asymptotic property of minimal rays, showing in particular that intersecting minimal geodesics as above can only occur as heteroclinic connections between pairs of homotopic closed minimal geodesics. A further application characterizes the boundary at infinity of the universal cover defined by Busemann functions. A third application shows that flat strips in the universal cover of a nonpositively curved surface are foliated by lifts of closed geodesics of a single homotopy class.     

Hyperbolic manifolds with convex boundary

Let (M,∂M) be a 3-manifold, which carries a hyperbolic metric with convex boundary. We consider the hyperbolic metrics on M such that the boundary is smooth and strictly convex. We show that the induced metrics on the boundary are exactly the metrics with curvature K>-1, and that the third fundamental forms of ∂M are exactly the metrics with curvature K<1, for which the closed geodesics which are contractible in M have length L>2π. Each is obtained exactly once. Other related results describe existence and uniqueness properties for other boundary conditions, when the metric which is achieved on ∂M is a linear combination of the first, second and third fundamental forms.     

Stable projective planes with Riemannian metrics

We consider the question whether the system of lines of a two-dimensional stable plane can be described as the system of geodesics of a Riemannian metric and vice versa; we present two results: A complete two-dimensional Riemannian manifold with the property that every two points are joined by a unique geodesic and its family of geodesics form a stable plane. On the other hand every stable projective plane whose lines are geodesics of a Riemannian metric is isometric to the real projective plane. Combining both results it follows that it is impossible to realize the lines of a non-desarguesian projective plane using the geodesics of a complete Riemannian manifold.     

The geometry at infinity of a hyperbolic Riemann surface of infinite type

We study geodesics on planar Riemann surfaces of infinite type having a single infinite end. Of particular interest is the class of geodesics that go out the infinite end in a most efficient manner. We investigate properties of these geodesics and relate them to the structure of the boundary of a Dirichlet polygon for a Fuchsian group representing the surface.      

The index growth and multiplicity of closed geodesics

In the recent paper [31] of Long and Duan (2009), we classified closed geodesics on Finsler manifolds into rational and irrational two families, and gave a complete understanding on the index growth properties of iterates of rational closed geodesics. This study yields that a rational closed geodesic cannot be the only closed geodesic on every irreversible or reversible (including Riemannian) Finsler sphere, and that there exist at least two distinct closed geodesics on every compact simply connected irreversible or reversible (including Riemannian) Finsler 3-dimensional manifold. In this paper, we study the index growth properties of irrational closed geodesics on Finsler manifolds. This study allows us to extend results in [31] of Long and Duan (2009) on rational, and in [12] of Duan and Long (2007), [39] of Rademacher (2010), and [40] of Rademacher (2008) on completely non-degenerate closed geodesics on spheres and CP² to every compact simply connected Finsler manifold. Then we prove the existence of at least two distinct closed geodesics on every compact simply connected irreversible or reversible (including Riemannian) Finsler 4-dimensional manifold.     

Geometry, topology and dynamics of geodesic flows on noncompact polygonal surfaces

We establish the background for the study of geodesics on noncompact polygonal surfaces. For illustration, we study the recurrence of geodesics on ?-periodic polygonal surfaces. We prove, in particular, that almost all geodesics on a topologically typical ?-periodic surface with a boundary are recurrent.     

Simple Curves on Surfaces

We study simple closed geodesics on a hyperbolic surface of genus g with b geodesic boundary components and c cusps. We show that the number of such geodesics of length at most L is of order L ^6g+2b+2c–6. This answers a long-standing open question.     

Geometry of the uniform infinite half‐planar quadrangulation

We give a new construction of the uniform infinite half‐planar quadrangulation with a general boundary (or UIHPQ), analogous to the construction of the UIPQ presented by Chassaing and Durhuus, which allows us to perform a detailed study of its geometry. We show that the process of distances to the root vertex read along the boundary contour of the UIHPQ evolves as a particularly simple Markov chain and converges to a pair of independent Bessel processes of dimension 5 in the scaling limit. We study the “pencil” of infinite geodesics issued from the root vertex as reported by Curien, Ménard, and Miermont and prove that it induces a decomposition of the UIHPQ into 3 independent submaps. We are also able to prove that balls of large radius around the root are on average 7/9 times as large as those in the UIPQ, both in the UIHPQ and in the UIHPQ with a simple boundary; this fact we use in a companion paper to study self‐avoiding walks on large quadrangulations.     

Insecurity for compact surfaces of positive genus

A pair of points in a riemannian manifold M is secure if the geodesics between the points can be blocked by a finite number of point obstacles; otherwise the pair of points is insecure. A manifold is secure if all pairs of points in M are secure. A manifold is insecure if there exists an insecure point pair, and totally insecure if all point pairs are insecure. Compact, flat manifolds are secure. A standing conjecture says that these are the only secure, compact riemannian manifolds. We prove this for surfaces of genus greater than zero. We also prove that a closed surface of genus greater than one with any riemannian metric and a closed surface of genus one with generic metric are totally insecure.     

Schottky Uniformizations of Genus 6 Riemann Surfaces Admitting A5 as Group of Automorphisms

In this note we construct a 1-complex dimensional family of (marked) Schottky groups of genus 6 with the property that every closed Riemann surface of genus 6 admitting the group A₅ as conformal group of automorphisms is uniformized by one of these Schottky groups. In the algebraic limit closure of this family we describe three noded Schottky groups uniformizing the three boundary points of the pencil described by González-Aguilera and Rodriguez. We are able to find a very particular Riemann surface of genus 6 which is a (local) extremal for a maximal set of homologically independent simple closed geodesics. We observe that it is not Wimann's curve, the only Riemann surface of genus 6 with S₅ as group of conformal automorphisms. The Schottky uniformizations permit us to compute a reducible symplectic representation of A₅.     

Length spectra and<Emphasis Type="Italic">p</Emphasis>-spectra of compact flat manifolds

We compare and contrast various length vs Laplace spectra of compact flat Riemannian manifolds. As a major consequence we produce the first examples of pairs of closed manifolds that are isospectral on p-forms for some p ≠ 0, but have different weak length spectrum. For instance, we give a pair of 4-dimensional manifolds that are isospectral on p-forms for p = 1, 3and we exhibit a length of a closed geodesic that occurs in one manifold but cannot occur in the other. We also exhibit examples of this kind having different injectivity radius and different first eigenvalue of the Laplace spectrum on functions. These results follow from a method that uses integral roots of the Krawtchouk polynomials. We prove a Poisson summation formula relating the p-eigenvalue spectrum with the lengths of closed geodesics. As a consequence we show that the Laplace spectrum on functions determines the lengths of closed geodesics and, by an example, that it does not determine the complex lengths. Furthermore we show that orientability is an audible property for closed flat manifolds. We give a variety of examples, for instance, a pair of manifolds isospectral on functions (resp. Sunada isospectral) with different multiplicities of length of closed geodesies and a pair with the same multiplicities of complex lengths of closed geodesies and not isospectral on p-forms for any p, or else isospectral on p-forms for only one value of p ≠ 0.     

Closed geodesics on pairs of pants

We study the non-simple closed geodesics of the Riemann surfaces of signature (0, 3). In the aim of classifying them, we define one parameter: the number of strings. We show that for a given number of strings, a minimal geodesic exists; we then give its representation and its length which depends on the boundary geodesics.     

Boundary Control of PDEs via Curvature Flows: the View from the Boundary, II

We describe some results on the exact boundary controllability of the wave equation on an orientable two-dimensional Riemannian manifold with nonempty boundary. If the boundary has positive geodesic curvature, we show that the problem is controllable in finite time if (and only if) there are no closed geodesics in the interior of the manifold. This is done by solving a parabolic problem to construct a convex function. We exhibit an example for which control from a subset of the boundary is possible, but cannot be proved by means of convex functions. We also describe a numerical implementation of this method.