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Harmonic maps are critical
points of the energy functional for maps between Riemannian manifolds. In this
paper we study the heat equation for harmonic maps from a non-compact manifold
M into N. We show that if the target manifold N is compact and has non-positive
sectional curvature, and if the initial map has finite total energy, then there exists a
solution u(x,t) : M × [0,∞) → N and a sequence tj →∞, such that u(⋅,tj)
converges on compact subsets of M to a harmonic from M into N. We also
obtain some basic properties of the solution u(x,t). In particular, we prove a
uniqueness theorem for the solution and a monotonicity theorem for the energy
functional.
Eells and Sampson proved that if (M,g) and (N,g′) are compact Riemannian
manifolds, (N,g′) has non-positive sectional curvature, then any smooth
map h : M → N is homotopic to a smooth harmonic map. They established
the existence of a solution u(x,t) : M × [0,∞) → N, of (1.1) in §1, and
showed that there exists tj →∞, such that u(⋅,tj) converges to a smooth
harmonic map from M into N. Schoen and Yau showed that if M is complete
non-compact and if h : M → N has finite energy, then h is homotopic on
any compact subsets of M to a harmonic map. Their method is based on
Hamilton’s results on harmonic maps from a manifold with boundary. By studying
the heat equation directly, we recovered the result of Schoen and Yau. We
believe the basic properties of solutions of the heat equation established in
this paper will be useful in the study of harmonic maps on non-compact
manifolds.
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