Eigenvalues of the Laplacian on forms

TLDR: Cheeger as mentioned in this paper showed that for a geodesic ball of radius smaller than the radius of injectivity, eigenvalues of the Laplacian Ap can be estimated from above and below in terms of bounds of sectional curvature.

Abstract: Some bounds for eigenvalues of the Laplace operator acting on forms on a compact Riemannian manifold are derived. In case of manifolds without boundary we give upper bounds in terms of the curvature, its covariant derivative and the injectivity radius. For a small geodesic ball upper and lower bounds of eigenvalues in terms of bounds of sectional curvature are given. In (2) Cheng proves a beautiful comparison theorem for the first eigenvalue of the Laplacian Ao on functions for a geodesic ball in a Riemannian manifold, and derives as a consequence, upper bounds for higher eigenvalues of the Laplacian on functions for a compact manifold. These upper bounds are derived by taking the first eigenfunctions with Dirichlet boundary conditions for small balls, extending them by zero to the whole manifold, and estimating the Rayleigh-Ritz quotient of an appropriate linear combination. The same procedure cannot be applied to forms of positive degree since an eigenfunction for a ball satisfying either absolute or relative boundary conditions will not be in the Sobolev space 771 when extended by zero to the whole manifold. In this note a modification of the method of Eichhorn (4) is used to prove that, for certain forms on a ball which vanish on the boundary, it is possible to estimate the Rayleigh-Ritz quotient in terms of geometric quantities. These forms are in H1 when extended by zero and Cheng's argument applied to them gives upper bounds for eigenvalues on a closed manifold. Our result is much less elegant than Cheng's theorem. In the first place we have to require that the geodesic ball is contained within the cut locus. Hence, all estimates of higher eigenvalues depend on the injectivity radius. Secondly, Cheng obtained explicit estimates of eigenvalues of the Laplacian Ao in terms of geometric quantities (cf. (2, Corollaries 2.2, 2.3, Theorem 2.4)), whereas we only say which geometric quantities determine the bounds of eigenvalues but give no estimates of the actual bounds. Explicit estimates could be derived from our method, but the constants are so complicated that we were unable to obtain any useful information from them. It is an interesting question, whether all of the geometric quantities which appear in our estimates (sectional curvature, injectivity radius, the bounds for the Kern tensor R^) are really necessary. We do not know the answer for closed manifolds. However, in §3 we show, following a suggestion of J. Cheeger, that for a geodesic ball of radius smaller than the radius of injectivity, eigenvalues of the Laplacian Ap can be estimated from above and below in terms of bounds of sectional curvature. It is a pleasure to thank J. Cheeger for suggestions which led to improvement of this paper.