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Index and total curvature of complete minimal surfaces. (English) Zbl 0592.53006
Geometric measure theory and the calculus of variations, Proc. Summer Inst., Arcata/Calif. 1984, Proc. Symp. Pure Math. 44, 207-211 (1986).
[For the entire collection see Zbl 0577.00014.]
Let $$M\subset R^ 3$$ be a complete minimal surface. M is said to have index k if the Jacobi operator $$L=-\Delta +2K$$ has at most k negative eigenvalues on any compact subset of M; here $$\Delta$$ is the Laplacian and K is the Gaussian curvature of the induced metric of M. A proof of the following result is presented; M has finite total curvature if and only if M has finite index.
That finite index implies finite total curvature is already contained in a paper by the author and H. B. Lawson jun. [Proc. Symp. Pure Math. 44, 213-237 (1986; see the preceding review)]. To prove the other half of the theorem, the author uses Osserman’s theorem that a complete minimal surface of finite total curvature is conformally equivalent to a compact Riemann surface $$\hat M$$ minus a finite number of points and that the Gauss map $$G: M\to S^ 2$$ extends to a branched conformal map from $$\hat M$$ to $$S^ 2$$. Let $$\hat M$$ denote M with the (singular) metric induced by G, and let $$\hat L=-{\hat \Delta}-2$$ be the Jacobi operator in $$\hat M.$$ The author proves that $$\hat L$$ has finite index, and this completes the proof. A further independent proof of the above result has been presented by D. Fischer-Colbrie [Invent. Math. 82, 121-132 (1985; Zbl 0573.53038)]. Fischer-Colbrie’s proof is self-contained and does not depend on the above quoted paper of the author and Lawson.
Reviewer: M.do Carmo

##### MSC:
 53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature 53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)