zbMATH — the first resource for mathematics

Eigenvalue estimates with applications to minimal surfaces. (English) Zbl 0594.58018
We study eigenvalue estimates of branched Riemannian coverings of compact manifolds. We prove that if \(\phi\) : \(M^ n\to N^ n\), is a branched Riemannian covering, and \(\{\mu_ i\}^{\infty}_{i=0}\) and \(\{\lambda_ i\}^{\infty}_{i=0}\) are the eigenvalues of the Laplace- Beltrami operator on M and N, respectively, then \[ \sum^{\infty}_{i=0}e^{-\mu_ it}\leq k\sum^{\infty}_{i=0}e^{- \lambda_ it}, \] for all positive t, 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 \({\mathbb{R}}^ 3\) 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.

58C40 Spectral theory; eigenvalue problems on manifolds
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
Full Text: DOI