Harmonic Maps on Smooth Metric Measure Spaces and on Riemannian Polyhedra

This thesis is a study of harmonic maps in two different settings. The first part is concerned with harmonic maps from smooth metric measure spaces to Riemannian manifolds. The second part is study of harmonic maps from Riemannian polyhedra to non-positively curved (locally) geodesic spaces in the sense of Alexandrov. The first part is organized as follows. We begin by defining a notion of harmonicity, and justify- ing the definition by checking it against pre-existing definitions and results in special cases. There are two main theorems in this section. The first is Theorem 0.1.1, which is the general- ization of the Shoen-Yau theorem [SY76] in our setting. The second is on the convergence of harmonic maps between Riemannian manifolds. Specifically we will show that if fi : Mi → N are a sequence of harmonic maps between Riemannian manifolds, and if the manifolds Mi converge to a smooth metric measure space M in the measured Gromov-Hausdorff topology, then the fi converge to a harmonic map f : M → N. This is the content of Theorem 0.1.2 In the second part, we prove Liouville-type theorems for harmonic maps under two different assumptions on the source space. First we prove the analogue of the Schoen-Yau theorem on a complete (smooth) pseudomanifolds with non-negative Ricci curvature. To this end we gen- eralize some Liouville- type theorems for subharmonic functions from [Yau76]. Then we study 2-parabolic admissible Riemannian polyhedra and prove vanishing results for subharmonic functions and harmonic maps on 2-parabolic pseudomanifolds.

Some regularity results for p-harmonic mappings between Riemannian manifolds

Changyu Guo

Let M be a C-2-smooth Riemannian manifold with boundary and N a complete C-2-smooth Riemannian manifold. We show that each stationary p-harmonic mapping u: M -> N, whose image lies in a compact subset of N, is locally C-1,C-alpha for some alpha is an element of (0, 1), provided that N is simply connected and has non-positive sectional curvature. We also prove similar results for minimizing p-harmonic mappings with image being contained in a regular geodesic ball. Moreover, when M has non-negative Ricci curvature and N is simply connected with non-positive sectional curvature, we deduce a gradient estimate for C-1-smooth weakly p-harmonic mappings from which follows a Liouville-type theorem in the same setting. (C) 2019 Elsevier Ltd. All rights reserved.

2019

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

Homogeneous Sobolev Metric of Order One on Diffeomorphism Groups on Real Line

Martin Bauer, Martins Bruveris

In this article we study Sobolev metrics of order one on diffeomorphism groups on the real line. We prove that the space equipped with the homogeneous Sobolev metric of order one is a flat space in the sense of Riemannian geometry, as it is isometric to an open subset of a mapping space equipped with the flat -metric. Here denotes the extension of the group of all compactly supported, rapidly decreasing, or -diffeomorphisms, which allows for a shift toward infinity. Surprisingly, on the non-extended group the Levi-Civita connection does not exist. In particular, this result provides an analytic solution formula for the corresponding geodesic equation, the non-periodic Hunter-Saxton (HS) equation. In addition, we show that one can obtain a similar result for the two-component HS equation and discuss the case of the non-homogeneous Sobolev one metric, which is related to the Camassa-Holm equation.

Springer Verlag2014

Optimal control of geodesics in Riemannian manifolds

Roland Rozsnyo

EPFL2007