Another Report on Harmonic Maps
نویسنده
چکیده
(1.1) Some of the main results described in [Report] are the following (in rough terms; notations and precise references will be given below): (1) A map (f>:{M,g)-+(N,h) between Riemannian manifolds which is continuous and of class L\ is harmonic if and only if it is a critical point of the energy functional. (2) Let (M, g) and (N, h) be compact, and <̂ 0: (M, g) -> (N, h) a map. Then ^0 can be deformed to a harmonic map with minimum energy in its homotopy class in the following cases: (a) Riem ' f t ^0; (b) dim M = 2 and n2(N) = 0. (3) Any map 0O: S m -> S can be deformed to a harmonic map provided m ^ 7. More generally, suitably restricted harmonic polynomial maps can be joined to provide harmonic maps between spheres. (4) The homotopy class of maps of degree 1 from the 2-torus T to the 2-sphere S has no harmonic representative, whatever Riemannian metrics are put on T and S. (5) If in (2) M has a smooth boundary, then various Dirichlet problems have solutions in case (a) and (b); and also when the boundary data is sufficiently small.
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