H-surface index formula

Abstract

We construct an additive index $\mathcal{I}$ on the set of compact “parts” of the set ${\mathcal{H}}_H(\Gamma )$ of “small” H-surfaces ($|H|<\frac{1}{2}$) that are spanned into a simple closed polygon $\Gamma \subset {\mathbb{R}}^3$ 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 ${\mathcal{H}}_H(\Gamma )$ is always 1, independent of $H$ and $\Gamma$. Moreover we compute that the Čech cohomology $\check{H}(\mathcal{P})$ of a part $\mathcal{P}$ that minimizes the H-surface functional ${\mathcal{E}}^H$ locally is non-trivial at most in degrees $0,\dots ,N-1$ and there even finitely generated, which implies the finiteness of the number of connected components of $\mathcal{P}$ in particular. Finally the index of such an “${\mathcal{E}}^H$-minimizing” part reveals to coincide with its Čech–Euler characteristic, which yields a variant of the mountain-pass-lemma.