Harmonic maps from 2-torus to 2-sphere and its heat flow

Abstract

Harmonic maps are the critical points of the Dirichlet energy functional for maps between two Riemannian manifolds. In this thesis, we study the corresponding heat flows (the negative gradient flows) from 2-torus to 2-sphere. In particular, we investigate the stability problem for minimizing harmonic maps within any given homotopy class. We show the stability result for the lineraized equation at a fixed steady state under a proper choice of parametrization for the perturbation term. We also consider a finite dimensional model problem of the gradient flow problem and prove its stability. Up to first order, it is analogous to the nonlinear problem but with the contracting part being finite-dimensional as well.