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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.

MSC:
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
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