On the optimality of J. Cheeger and P. Buser inequalities. (Q1417340)

Let \(\lambda_1(M,g)\) be the first non-zero eigenvalue of the scalar Laplacian for a closed \(n\) dimensional Riemannian manifold. Let \(h(M,g)\) be the Cheeger isoperimetric constant. The Cheeger inequality yields \(\lambda_1(M,g)\geq h(M,g)^2/4\). If \(Ric(M,g)\geq-(n-1)\delta^2g\), where \(Ric\) is the Ricci tensor, then the Buser inequality gives \[ \lambda_1(M,g)\leq c(n)(| \delta| h(M,g)+h(M,g)^2). \] The authors study the sharpness of these inequalities as the Cheeger constant becomes small. They show: Theorem 1. Let \(M\) be an \(n\) dimensional closed manifold. For any fixed \(\alpha\in[1,2]\) there exists a family of Riemannian metrics \(g_j\) on \(M\) and positive constants \(C_i\) so that (1) \(Ric(M,g_j)\geq-(n-1)\delta^2g_j\), (2) \(\lim_j h(M,g_j)=0\), and (3) \(C_1\leq\lambda_1(M,g_j)/h(m,g_j)^\alpha\leq C_2\). \smallbreak They also study the hyperbolic setting Theorem 2. For any \(n\geq2\) and for any \(\alpha\in[1,2]\), there exists a family of \(n\) dimensional hyperbolic manifolds \(M_j\) and positive constants \(D_i\) so that (1) \(\lim_jh(M_j)=0\) and (2) \(D_1\leq \lambda_1(M_i)/h(M_i)^\alpha\leq D_2\). \smallbreak Similar related questions on the combinatorial level for graphs are also discussed.