Harmonic maps on complete manifolds
In this article, we study harmonic maps between two complete non-compact manifolds M and N by a heat flow method. We find some new sufficient conditions to ensure the uniform convergence of the heat flow, and hence the existence of harmonic maps. Our conditions are: The Ricci curvature of M is bounded from below by a negative constant, M admits a positive Green's function, and ∫M G(x,y)|τ(h(y))|dVy is bounded on each compact subset. where τ(h(x)) is the tension field of the initial data h(x). Conditions (1) are somewhat sharp as is shown by examples in the paper.
Harmonic maps between complete, Heat flow method, Noncompact manifolds, Uniform convergence of heat flows
Chen, Wenxiong, and Congming Li. "Harmonic maps on complete manifolds." Discrete and Continuous Dynamical Systems 5, no. 4 (1999): 799.
Discrete and Continuous Dynamical Systems