Computing harmonic maps between Riemannian manifolds
Date/Time:22 Nov 2019 14:00Venue: S17 #04-04 SR3Speaker: Brice Loustau , TU DarmstatComputing harmonic maps between Riemannian manifoldsThe theory of harmonic maps between Riemannian manifolds goes back to the foundational work of Eells-Sampson and Hartman. Using the heat flow method, i.e. the gradient flow for the energy functional, they showed existence and uniqueness of harmonic maps between compact Riemannian manifolds when the target has negative sectional curvature. In order to compute such harmonic maps effectively, we discretize the theory by introducing meshes of the domain manifold, which consist of a geodesic triangulation with vertex and edge weights. By studying the convexity properties of the energy functional, we show that the discrete heat flow method converges to unique discrete harmonic maps. Moreover, under suitable assumptions, the discrete theory converges back to the smooth theory when taking finer and finer meshes. In particular, the smooth harmonic map can be obtained as the limit of discrete harmonic maps, equivalently as the double limit of the discrete heat flow when iterating both the time and space discretization steps. Time permits, we will also discuss center of mass methods to compute harmonic maps. We feature a concrete illustration of these methods withHarmony, a computer software with a graphical user interface that we developed in C++, whose main functionality is to numerically compute and display equivariant harmonic maps between hyperbolic disks.Add to calendar:
