AN UPPER BOUND FOR SMALL EIGENVALUES OF THE LAPLACIAN
Neil N. Katz · Osaka City University (Osaka City University) · 2003
M. Berger gave a curvature free upper bound for the first eigenvalue in terms of injectivity radius and dimension for a compact Riemannian manifold admitting a fixed point free, isometric involution [2].P. Bérard and G. Besson [1] extended this result to homogeneous and globally harmonic Riemannian manifolds.C. Croke [4] improved Berger's estimate for the Dirichlet problem with a bound in terms of convexity radius which gave as a corollary an upper bound for compact manifolds.In this note we give an estimate (Theorem 3) which is sharper than those mentioned above with the additional hypothesis of an upper curvature bound but without global hypothesis on the injectivity radius.Moreover, this estimate is sharp with equality holding only in the case of spheres of constant curvature and gives bounds for higher eigenvalues if the dimension is at least three.If the manifold is homeomorphic to certain -dimensional spherical space forms and the bounds on the injectivity radius and sectional curvature hold globally, then we can give an upper bound for the th eigenvalue (Theorem 4).A sharp lower bound for the sum of the reciprocals of the first three eigenvalues of 2 was made by Hersch [10] in terms of area alone.P. Yang and S.-T.Yau [17] generalized this estimate to compact surfaces in terms of genus and area.Both of these results give an upper bound for the first eigenvalue which is sharp in the case of spheres.P. Li and S.-T.Yau [12] reproduced this estimate by employing a conformal invariant, the conformal area.They were also able to give a sharp estimate in the case of the real projective plane and for the conformal class of the square, flat torus also only in terms of area.In higher dimensions their estimates require ( ) to be conformally equivalent to an immersed, minimal submanifold of the standard sphere of dimension ≥ .Examples of H. Urakawa [15] and J. Dodziuk [5] show that for dimension at least three, there does not exist an upper bound for the first eigenvalue in terms of volume alone.Other upper estimates for λ 1 involve either a lower bound on curvature [3], [7], or hold only for surfaces [6], [9], [10], [14].See [11] for a survey of eigenvalue bounds.We will use the following notation.The metric ball of radius at a point