H-surface index formula

We construct an additive index I on the set of compact “parts” of the set HH(Γ) of “small” H-surfaces (|H|<12) that are spanned into a simple closed polygon Γ⊂R3 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 HH(Γ) is always 1, independent of H and Γ. Moreover we compute that the Čech cohomology Hˇ(P) of a part P that minimizes the H-surface functional EH 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 “EH-minimizing” part reveals to coincide with its Čech-Euler characteristic, which yields a variant of the mountain-pass-lemma.