# H-surface index formula

### Ruben Jakob

Mathematisches Institut der Universität, Bonn, Germany

## Abstract

We construct an additive index $I$ on the set of compact “parts” of the set $H_{H}(Γ)$ of “small” H-surfaces ($∣H∣<21 $) that are spanned into a simple closed polygon $Γ⊂R_{3}$ with $N+3$ vertices ($N⩾1$) by a combination of Heinz' and Hildebrandt's examinations of H-surfaces and Dold's fixed point theory. We obtain that the index of $H_{H}(Γ)$ is always 1, independent of *H* and *Γ*. Moreover we compute that the Čech cohomology \( H\limits^{ˇ}(\mathscr{P}) \) of a part $P$ that minimizes the H-surface functional $E_{H}$ locally is non-trivial at most in degrees $0,…,N−1$ and there even finitely generated, which implies the finiteness of the number of connected components of $P$ in particular. Finally the index of such an “$E_{H}$-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