Eigenvalue estimates with applications to minimal surfaces

Abstract

We study eigenvalue estimates of branched Riemannian coverings of compact manifolds. We prove that if : M n -> N n is a branched Riemannian covering, and {/i/}Jio and {, }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 consequence 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.