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Lower bounds for the Morse index of complete minimal surfaces in Euclidean 3-space. (English) Zbl 0704.53006
Let M be a minimal surface in $$R^ 3$$. The index of M is the supremum of the indices of the Jacobi operator on relatively compact domains in M and is denoted by Ind(M). Let $$\Sigma$$ be a compact Riemann surface. For a nonconstant holomorphic map $$G: \Sigma \to S^ 2$$ and a subset $$A\subset \Sigma$$, set $$b(G,A)=\sum_{p\in A}b(G,p),$$ where b(G,p) is the branching order of G at p. The main result of the present paper is the following: Let M be a complete oriented nonplanar minimal surface in $$R^ 3$$ of genus g with finite total curvature and $$\tilde G: \tilde M\to S^ 2$$ the extended Gauss map, where $$\tilde M$$ is the compactified surface. Then the inequality $$Ind(M)\geq b(\tilde G,\tilde G^{-1}(S^ 1))+1-2g$$ holds for any great circle $$S^ 1$$ of $$S^ 2$$. As the corollary, the author obtains: Let M be a complete oriented minimal surface in $$R^ 3$$ of genus zero. Suppose that M is not one of the plane, the Enneper’s surface and the catenoid. Then Ind(M)$$\geq 3$$.
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.)