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Zbl 0553.53026
Kasue, Atsushi
On a lower bound for the first eigenvalue of the Laplace operator on a Riemannian manifold. (English)
[J] Ann. Sci. Éc. Norm. Supér. (4) 17, 31-44 (1984). ISSN 0012-9593

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 $\lambda\sb 1$, the diameter of M is d, and the in-radius of M is I. Theorem 1. If M is of class (R,$\Lambda)$ then $\lambda\sb 1\le L=$ 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 $\lambda\sb 1>A=$ 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. \par 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: $$ B=\pi\sp 2/(4d\sp 2)\quad if\quad K=0;\quad B=(m-1)\sp 2\vert K\vert /(4(1-\exp (-(m-1)\vert K\vert\sp{1/2}d/2))\sp 2)\quad if\quad K<0. $$ 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, {\it P. Li} and {\it S.-T. Yau} [Proc. Symp. Pure Math. 36, 205-239 (1980; Zbl 0441.58014)] obtain similar results using gradient estimates.
[R.Reilly]

MSC 2000:: *53C20 Riemannian manifolds (global)
58J50 Spectral problems; spectral geometry; scattering theory

Keywords: eigenvalue estimates; Busemann functions; Ricci curvature; mean curvature; eigenvalue of the Laplacian; Jacobi differential equation

Citations: Zbl 0518.53048; Zbl 0502.53034; Zbl 0441.58014

Cited in: Zbl 0615.53036 Zbl 0578.53029

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