Isoperimetric bounds for lower-order eigenvalues

We adopt the convention that each eigenvalue is repeated according to its multiplicity. An important issue in spectral geometry is to obtain good estimates for these and other eigenvalues in terms of the geometric data of the manifold M such as the volume, the diameter, the curvature, the isoperimetric constants, etc. See [1],[2],[10],[13],[31] for references. On the other hand, after the seminal works of Bleecker-Weiner [4] and Reilly [30], the following approach is developed: the manifold (M, g) is immersed isometrically into another Riemannian manifold. One then gets good estimates for λk(M), mostly for λ1(M), in termos of the extrinsic geometric quantities of M . See for example [4], [15], [16], [23], [24], [35], [37]. Especially relevant for us is the quoted work of Reilly [30], where he obtained the following remarkable isoperimetric inequality for the first positive eigenvalue λ1(M) in the case that M is embedded as a hypersurface bounding a domain Ω in R: λ1(M) ≤ n− 1 n2 · |M | 2 |Ω|2 . (1.1)