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On the Morse index of complete minimal surfaces in Euclidean space. (English) Zbl 0704.53007

Let M be a minimal surface in \(R^ n\). The index of M is the supremum of the indices of the Jacobi operator on relatively compact domains in M. Fischer-Colbrie [Invent. Math. 82, 121-132 (1985; Zbl 0573.53038)], R. Gulliver and H. B. Lawson jun. [Geometric measure theory and the calculus of variations, Proc. Summer Inst., Arcata/Calif. 1984, Proc. Symp. Pure Math. 44, 213-237 (1986; Zbl 0592.53005)] proved that a complete oriented minimal surface in \(R^ 3\) has finite index if and only if it has finite total curvature. In the present paper, it is shown that a complete minimal surface in \(R^ n\) with finite total curvature has finite index. When \(n=4\), the author proves the converse: Let M be a complete oriented minimal surface in \(R^ 4\) and conformally equivalent to a compact Riemann surface with finitely many punctures. Suppose that M has finite index and that the Gauss map of M is degenerate. Then M is of finite total curvature or a holomorphic curve with respect to some orthogonal complex structure on \(R^ 4\).
Reviewer: T.Ishihara

MSC:

53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
58E12 Variational problems concerning minimal surfaces (problems in two independent variables)
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