H-surface index formula

  • Ruben Jakob

    Mathematisches Institut der Universität, Bonn, Germany


We construct an additive index on the set of compact “parts” of the set of “small” H-surfaces () that are spanned into a simple closed polygon with vertices () by a combination of Heinz' and Hildebrandt's examinations of H-surfaces and Dold's fixed point theory. We obtain that the index of is always 1, independent of H and Γ. Moreover we compute that the Čech cohomology \( H\limits^{ˇ}(\mathscr{P}) \) of a part that minimizes the H-surface functional locally is non-trivial at most in degrees and there even finitely generated, which implies the finiteness of the number of connected components of in particular. Finally the index of such an “-minimizing” part reveals to coincide with its Čech–Euler characteristic, which yields a variant of the mountain-pass-lemma.

Cite this article

Ruben Jakob, H-surface index formula. Ann. Inst. H. Poincaré Anal. Non Linéaire 22 (2005), no. 5, pp. 557–578

DOI 10.1016/J.ANIHPC.2004.10.008