Isoperimetric control of the spectrum of a compact hypersurface (Q2861472)

Let \((\Sigma ,g)\) be a compact Riemannian manifold of dimension \(n\geq 2.\) The spectrum \(\{\lambda _{i}\mid 0=\lambda _{1}\leq \lambda _{2}\leq \dots \nearrow \infty \}\) of eigenvalues of the Laplace-Beltrami operator are global Riemannian invariants which relate to geometric properties of the manifold. The study of the spectrum here is that of finding bounds on the eigenvalues of the Laplacian on \(\Sigma \) as a hypersurface bounding a domain \(\Omega \) in Euclidean or more general ambient spaces. Given a bounded domain \(\Omega \subset \mathbb{R}^{n+1}\) with smooth boundary \(\Sigma \), the authors obtain bounds for each eigenvalue which depend on the isoperimetric ratio \(I(\Omega)\equiv \frac{|\Sigma|}{|\Omega|^{ \frac{n}{n+1}}},\) where \(|\Sigma|\) and \(|\Omega|\) are the Riemannian \(n\) and \(n+1\) volumes respectively of \(\Sigma \) and \(\Omega .\) In particular they obtain that \(\lambda _{k}|\Sigma|^{\frac{2}{n}}\leq \gamma _{n}I(\Omega)^{1+\frac{2}{n}}k^{\frac{2}{n}}\), where \(\gamma _{n}\) is a function of \(n\) and the volume of the unit ball in \(\mathbb{R}^{n+1}\). The authors give an interesting interpretation of the result: when \((\Sigma ,g)\) is an \(n\)-dimensional Riemannian manifold isometrically embedded by \(\varphi :\Sigma \rightarrow \mathbb{R}^{n+1}\) with \(|\Sigma|=1\), then the domain \(\Omega \) bounded by the hypersurface \(\varphi (\Sigma)\) satisfies \(|\Omega|^{ \frac{n+2}{n+1}}\leq \gamma _{n}\frac{k^{\frac{2}{n}}}{\lambda _{k}}\), \( \forall k\geq 2\). More generally, suppose \(M\) is a complete \((n+1)\)-dimensional Riemannian manifold with Ricci curvature bounded below and \(\Omega \subset M\) is a bounded domain with smooth boundary \( \Sigma \), the authors obtain an upper bound for each eigenvalue depending on \(k,\) \(n\) and isoperimetric ratios. The authors then generalize this result so that the curvature assumption is replaced by metric conditions.