Existence Of Harmonic Maps And The Heat Flow Equation

Harmonic maps are one kind of important objects in geometry analyses. The existence problem of harmonic maps is a very interesting but very difficult problem. One of the most important and interesting aspects of the study of the existence problem of harmonic maps is the existence problem between two manifolds. In generally, we only consider the following two cases. One case is the two manifolds have no boundary; the other case is the two manifolds have a boundary. About the manifolds have a boundary, if we use the definition of harmonic maps, it's fail to apply the direct method in variation techniques to find out the question. This method is only operated in some special manifolds. For example, in S1, we can use this method [2]. The reason is that according to S1, it's harmonic maps are closed geodesics. The equation for harmonic maps is essentially a system of nonlinear partial differential equations, as oppose to that the defining equation for geodesics is a system of linear ordinary equation in the tangent bundle TN of N. In order to find out this question, Eells and Sampson introduced a technique called the heat flow method [1]. It is considered to be the most fundamental among existence theorems for harmonic maps.This paper pays attention to descript the ideas of Eells and Sampson, and explain Eells and Sampson's proof in 1964 about their theorem in details by myself. The theorem which the paper wants to proof is as follows:Theorem4.1. ( Eells-Sampson ) Let (M , g) and ( N, h) be compact Riemannian manifolds. Suppose that N has nonpositive section curvature. Then for any f∈C∞(M ,N), there is a harmonic map u∞: Mâ†'N free-homotopic to f.This paper has two parts. In the first part, the paper concludes the equation for harmonic maps by the definition of harmonic maps, and takes two examples of the harmonic maps between two special manifolds,; in the second part, firstly, this paper introduces the existence problem of harmonic maps between two manifolds which have a boundary. Secondly, the paper pays attention to descript the ideas of Eells and Sampson—the heat flow method, namely, the existence problem of harmonic maps is reduced to the initial value problem of the parabolic equation for harmonic maps, and proofs the initial value problem by two aspects which are the existence of solutions and the converging of solutions, thirdly, the paper supposes the existence of solutions, and proofs he converging of solutions by Weitzenb ock formula, finally, the paper proofs the existence of solutions.During the proof of the existence of solutions, the paper also has two parts.The first part is the existence of local time-dependent solutions. The paper reduces the initial value problem of the parabolic equation to the initial value problem of the vector value functions by the imbedding theorem [7], then the paper proofs the initial value problem of the vector value functions by the inverse function theorem in Banach space. So the paper proofs the existence of local time-dependent solutions.The second part is the existence of global time-dependent solutions. The paper supposes N has nonpositive section curvature, and gets the estimate of u by Weitzenb ock formula. Then, the paper gets the uniformly estimate of u by the Schauder estimate for linear parabolic partical differential equations, so we can proofs the existence of global time-dependent solutions.Here, we completely proof Eells and Sampson's theorem.Similarly, we can study the existence problem of analytic maps between complex manifolds. Especially since analytic maps between Kahler manifolds are harmonic maps. Indeed, as an application of the theorem of Eells and Sampson, Siu [10],[11] proved a strong rigidity theorem regarding complex structure in Kahler manifolds of negative curvature in 1980.