an:03956033
Zbl 0594.58018
Tysk, Johan
Eigenvalue estimates with applications to minimal surfaces
EN
Pac. J. Math. 128, No. 2, 361-366 (1987).
00165350
1987
j
58C40 53C42 58J50
branched Riemannian covering; eigenvalues; Laplace-Beltrami operator; minimal surface; heat kernel
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.