Morse Theory for Harmonic Maps
In the previous paper [Ch1], we studied the Minimax Principle as well as the Ljusternik–Schnirelman category theory for harmonic maps with prescribed boundary data defined on Riemann surfaces by the heat flow method. In this paper, we shall continue our study on Morse theory. Our main results are the Morse inequalities (Theorem 1) for isolated harmonic maps, and the Morse handle body decomposition for nondegenerate harmonic maps (Theorem 2). These results are extensions of the work of K. Uhlenbeck [U1], where the harmonic maps are defined on manifolds without boundary, and are all assumed to be nondegenerate. Our method is based on the heat flow by which the deformation is constructed. In contrast to the perturbation method developed by K. Uhlenbeck [U1], our approach seems more direct than hers.
KeywordsBetti Number Morse Index Morse Theory Minimax Principle Morse Decomposition
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