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On a lower bound for the first eigenvalue of the Laplace operator on a Riemannian manifold. (English) Zbl 0553.53026

Let M be an m-dimensional compact connected Riemannian manifold with smooth boundary \(\partial M\). Say that M is ”of class (R,\(\Lambda)\)” if the Ricci curvature of M is bounded below by (m-1)R and the mean curvature H of \(\partial M\) is bounded above by \(\Lambda\). (Signs are chosen so \(H<0\) if \(\partial M\) is convex.) The first (Dirichlet) eigenvalue of the Laplacian on M is \(\lambda_ 1\), the diameter of M is d, and the in-radius of M is I. Theorem 1. If M is of class (R,\(\Lambda)\) then \(\lambda_ 1\leq L=\) constant depending only on m, R, \(\Lambda\), and I. This bound is achieved precisely when M is a ”model space of class (R,\(\Lambda)\)”. Theorem 2: Suppose M is a compact domain in a complete noncompact Riemannian manifold N (\(\partial N\) empty). (1) If (m-1)R is a nonpositive lower bound on the Ricci curvature of N, then \(\lambda_ 1>A=\) constant depending only on m, R, and d. (2) If, instead, the sectional curvature is bounded from above by a nonpositive constant K and N admits a concave function without a maximum, then \(\lambda >B=\) constant depending only on K and d.
The model spaces of class (R,\(\Lambda)\) and the constant L are described in terms of a Jacobi differential equation. The constants A and B are given explicitly. For instance: \[ B=\pi^ 2/(4d^ 2)\quad if\quad K=0;\quad B=(m-1)^ 2| K| /(4(1-\exp (-(m-1)| K|^{1/2}d/2))^ 2)\quad if\quad K<0. \] The author uses facts and Jacobi equation techniques from his earlier papers [Jap. J. Math., New Ser. 18, 309-341 (1982; Zbl 0518.53048) and J. Math. Soc. Japan 35, 117- 131 (1983; Zbl 0502.53034)]; in contrast, P. Li and S.-T. Yau [Proc. Symp. Pure Math. 36, 205-239 (1980; Zbl 0441.58014)] obtain similar results using gradient estimates.
Reviewer: R.Reilly

MSC:

53C20 Global Riemannian geometry, including pinching
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
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References:

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