A Morse theory for annulus-type minimal surfaces. (English) Zbl 0605.58018

Struwe, Michael

J. Reine Angew. Math. 368, 1-27 (1986).

M. Morse and C. Tompkins [Duke Math. J. 8, 350-375 (1941; Zbl 0025.40902)] extended the critical point theory developed by Morse to minimal surfaces of higher topological structure. A new view in this problem was introduced by R. Böhme and A. J. Tromba [Ann. Math., II. Ser. 113, 447-499 (1981; Zbl 0482.58010)], who investigated the global structure of the set of all minimal surfaces spanned by all smooth curves in \({\mathbb{R}}^ N\), \(N\geq 3\). The author of the paper under review has developed an independent variational approach to Plateau’s problem [J. Reine Angew. Math. 349, 1-23 (1984; Zbl 0521.49028)] which in particular implied all the Morse inequalities for a generic boundary curve in \({\mathbb{R}}^ N\), \(N\geq 4\). In the present paper the author conveys this method to Plateau’s problem for minimal surfaces of genus zero of higher connectivity. The author is concerned with laying the functional analytic frame, derives mountain-pass-type existence results for unstable minimal surfaces of annulus type and works out a complete Morse theory for this problem in the non-degenerate case. He considers for simplicity only the case of two boundary contours, however its methods seem to generalize to arbitrary connectivity.

Reviewer: F.Klepp

MSC:

58E05	Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
58E12	Variational problems concerning minimal surfaces (problems in two independent variables)

Keywords:

critical point theory; Plateau’s problem; Morse inequalities; minimal surfaces; mountain-pass-type existence results

Citations:

Zbl 0025.40902; Zbl 0482.58010; Zbl 0521.49028