Capacities, Systoles and Jacobians of Riemann Surfaces

To any compact Riemann surface of genus g one may assign a principally polarized abelian variety (PPAV) of dimension g, the Jacobian of the Riemann surface. The Jacobian is a complex torus and we call a Gram matrix of the lattice of a Jacobian a period Gram matrix. The aim of this thesis is to contribute to the Schottky problem, which is to discern the Jacobians among the PPAVs. Buser and Sarnak approached this problem by means of a geometric invariant, the first successive minimum. They showed that the square of the first successive minimum, the squared norm of the shortest non-zero vector, in the lattice of a Jacobian of a Riemann surface of genus g is bounded from above by log(4g), whereas it can be of order g for the lattice of a PPAV of dimension g. The main goal of this work was to improve this result and to get insight into the connection between the geometry of a compact Riemann surface that is given in hyperbolic geometric terms, and the geometry of its Jacobian. We show the following general findings: For a hyperelliptic surface the first successive minimum is bounded from above by a universal constant. The square of the second successive minimum of the Jacobian of a Riemann surface of genus g is equally of order log(g). We provide refined upper bounds on the consecutive successive minima if the surface contains several disjoint small simple closed geodesics and a lower bound for the norm of certain lattice vectors of the Jacobian, if the surface contains small non-separating simple closed geodesics. If the concrete geometry of the Riemann surface is known, more precise statements can be made. In this case we obtain theoretical and practical estimates on all entries of the period Gram matrix. Here we establish upper and lower bounds based on the geometry of the cut locus of simple closed geodesics and also on the geometry of Q-pieces. In addition the following two results have been obtained: First, an improved lower bound for the maximum value of the norm of the shortest non-zero lattice vector among all PPAVs in even dimensions. This follows from an averaging method from the geometry of numbers applied to a family of symmetric PPAVs. Second, a new proof for a lower bound on the number of homotopically distinct geodesic loops, whose length is smaller than a fixed constant. This lower bound applies not only to geodesic loops on Riemann surfaces, but on arbitrary manifolds of non-positive curvature.

Optimal control of geodesics in Riemannian manifolds

Roland Rozsnyo

The aim of this dissertation is to solve numerically the following problem, denoted by P : given a Riemannian manifold and two points a and b belonging to that manifold, find a tangent vector T at a, such that expa(T) = b, assuming that T exists. This problem is set under an optimal control formulation, which requires the definition of an objective function and a space of control, the choice of a method for the calculation of the descent direction of that function in the space of control and the use of an optimization algorithm to find its minimum, which corresponds to the solution of the original problem by construction. Several techniques are necessary to be put together, coming from the fields of geometry, numerical analysis and optimization. The first part will concern a recalling of the mathematical context in which this formulation takes place. The general principles of optimal control will also be given. In the second part, we will present an intrinsic formulation of the optimal control problem associated to P, based on Jacobi fields, which will play the role of the so called adjoint state. This derivation leads to necessary optimality conditions. We will illustrate explicitly that formulation by treating the specific case of Riemannian manifolds with constant sectional curvature. Then, we will derive the optimal control problem in coordinates, not only to check the intrinsic formulation but also to reveal how it is hidden behind the expressions in coordinates. Their use reveals some quantities whose interpretation may be given this way. Moreover, we will show that more possibilities exist to chose the cost function and the control space in coordinates. In a second step, an alternative approach will consider the Hamiltonian formulation of geodesics. This is an incursion into symplectic geometry. We will then reformulate the Riemannian optimal control problem in its Hamiltonian version. In the third part, the numerical methods used for solving P will be presented. The discretization imposes the definition of new discrete optimal control problems. The technique shows that the discrete adjoint state equation strongly depends on the numerical scheme used to solve the direct problem. We will give a collection of numerical computations in the specific case of parametric piece of surfaces, where the surface can be defined by one or several Bézier patches, each one corresponding to a chart, which is representative of a Riemannian manifold. We will compare the different numerical approaches. The last but one part will be devoted to the interesting application of wooden roof building, where the structure is made of wooden boards, with geodesic trajectories on the designed piece of surface. The Geos (Geodesic solver) software has been developed for that purpose. After having introduced some specific numerical methods used in the code, we present the Geos application interface (AI) developed as a tool for the conception of such a roof. We then show an existing wooden structure built according to that mean. Finally, we will summarize the results of our research and discuss future possible prospects.

EPFL2007

Trace coordinates of Teichmüller spaces of Riemann surfaces

Thomas Gauglhofer

Using an algebraic formalism based on matrices in SL(2,R), we explicitly give the Teichmüller spaces of Riemann surfaces of signature (0,4) (X pieces), (1,2) ("Fish" pieces) and (2,0) in trace coordinates. The approach, based upon gluing together two building blocks (Q and Y pieces), is then extended to tree-like pants decomposition for higher signatures (g,n) and limit cases such as surfaces with cusps or cone-like singularities. Given the Teichmüller spaces, we establish a set of generators of their modular groups for signatures (0,4), (1,2) and (2,0) in trace coordinates using transformations acting separately on the building blocks and an algorithm on dividing geodesics. The fact that these generators act particularly nice in trace coordinates gives further motivation to this choice (rather then the one of Fenchel-Nielsen coordinates). This allows us to solve the Riemann moduli problem for X pieces, "Fish" pieces and surfaces of genus 2; i.e. to give the moduli spaces as the fundamental domains for the action of the modular groups on the Teichmüller spaces. In this context, we also give an algorithm deciding whether two Riemann surfaces of signatures (0,4), (1,2) or (2,0) given by points in the Teichmüller space are isometric or not. As a consequence, we show the following two results concerning simple closed geodesics: On any purely hyperbolic Riemann surface (containing neither cusps nor cone-like singularities), the longest of two simple closed geodesics that intersect one another n times is of length at least ln, a sharp constant independent of the surface. We explicitly give ln for n = 1,2,3 and study its behaviour when n goes to infinity. X pieces are spectrally rigid with respect to the length spectrum of simple closed geodesics.

EPFL2006

Finite Element Method on Riemann Surfaces and Applications to the Laplacian Spectrum

Marc Maintrot

The objective of this PhD thesis is the approximate computation of the solutions of the Spectral Problem associated with the Laplace operator on a compact Riemann surface without boundaries. A Riemann surface can be seen as a gluing of portions of the Hyperbolic Plane made with suitable conditions to obtain a 2 dimensional manifold. The solutions of the Spectral Problem associated with the Laplace operator are to be understood as the eigenfunctions defined on the surface and their corresponding eigenvalues. This work is separated into two parts: the first part describes the method used to approximate the eigenvalues and eigenfunctions, the second focuses on the design of a program to compute these approximations. The approximation method is inspired by the Finite Element Method (FEM), in that it relies on the variational expression of the Spectral Problem and the definition of a finite subspace of functions in which the approximated eigenvalues and eigenfunctions are computed. However, it differs from the FEM in that it removes the euclidian basis of the FEM and is invariant under the isometries of the Hyperbolic Plane. To ful fill this objective, we begin by geodesically triangulating the surface as regularly as possible. This hyperbolic triangulation allows us to de ne the finite subspace of functions by using the concept of barycentric coordinates associated with each vertex of the triangulation (idea introduced by Whitney and taken up by Dodziuk). We then prove that the approximated solutions convergence to the exact ones when the diameter of the triangulation decreases, as well as the order of convergence. The program is a practical application of the theoretical work and allows the computation of the approximated eigenfunctions and eigenvalues.

EPFL2012

Optimal control of geodesics in Riemannian manifolds

Roland Rozsnyo

EPFL2007