Morse Theory for Harmonic Maps

  • Kung-Ching Chang
Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 4)


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.


Betti Number Morse Index Morse Theory Minimax Principle Morse Decomposition 
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  1. [Ch1]
    K.C. Chang, Heat flow and boundary value problem for harmonicmaps,Ann. Inst. Henri Poincaré, Anal. nonlinéaire 6 (1989), 363–395.Google Scholar
  2. [Ch2]
    K.C. Chang, Infinite dimensional Morse theory and its applications Université de Montréal (1985).Google Scholar
  3. [ELI]
    J. Eells, L. L.maire, Selected topics in harmonic maps, CBMS Reg. Conf. Series no. 50, (1983).Google Scholar
  4. [U1]
    K. Uhlenbeck, Morse theory by perturbation methods with applications to harmonic maps, TAMS (1981).Google Scholar

Copyright information

© Springer Science+Business Media New York 1990

Authors and Affiliations

  • Kung-Ching Chang
    • 1
  1. 1.Peking UniversityBeijingChina

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