Eigenvalue estimates with applications to minimal surfaces

Johan Tysk
1987 Pacific Journal of Mathematics  
We study eigenvalue estimates of branched Riemannian coverings of compact manifolds. We prove that if φ : M n -> N n is a branched Riemannian covering, and {/i/}Jio and {λ, }°L 0 are the eigenvalues of the Laplace-Beltrami operator on M and N, respectively, then for all positive /, where k is the number of sheets of the covering. As one application of this estimate we show that the index of a minimal oriented surface in R 3 is bounded by a constant multiple of the total curvature. Another
more » ... uence of our estimate is that the index of a closed oriented minimal surface in a flat three-dimensional torus is bounded by a constant multiple of the degree of the Gauss map. The Supporting Institutions listed above contribute to the cost of publication of this Journal, but they are not owners or publishers and have no responsibility for its content or policies. Mathematical papers intended for publication in the Pacific Journal of Mathematics should be in typed form or offset-reproduced (not dittoed), double spaced with large margins. Please do not use built up fractions in the text of the manuscript. However, you may use them in the displayed equations. Underline Greek letters in red, German in green, and script in blue. The first paragraph must be capable of being used separately as a synopsis of the entire paper. In particular it should contain no bibliographic references. Please propose a heading for the odd numbered pages of less than 35 characters. Manuscripts, in triplicate, may be sent to any one of the editors. Please classify according to the scheme of Math. Reviews, Index to Vol.
doi:10.2140/pjm.1987.128.361 fatcat:2qolo3m4yjdmxidppf2z2nm7b4