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.

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