An eigenvalue bound for compact manifolds with varying curvature (Q1826145)

Let \(\rho\) be the Ricci tensor of a compact m dimensional manifold M and let \(\lambda\) (x) be the lowest eigenvalue of \(\rho\) ; assume \(\lambda\) (x)\(\geq 0\) and \(\lambda\) \(\not\equiv 0\). Let \(\lambda_ 1\) be the lowest non-zero eigenvalue of the Laplacian \(\Delta =d\delta\). If \(\lambda\) (x)\(\geq \alpha\), Lichnerowicz showed Theorem 1 (Lichnerowicz): \(\lambda_ 1\geq m\alpha /(m-1).\) Let \(W(x)=2(\alpha -\alpha \lambda (x))\) where \(\alpha\) is constant and \(0\leq W\leq 2\alpha\). Let \(|_{p,q}\) be the norm of an operator from \(L^ p\) to \(L^ q\) and let \(c=| W(2\alpha +\Delta)^{- 1}|_{1,1};\quad 0\leq c<1.\) Theorem 2: The smallest non-zero eigenvalue \(\lambda_ 1\) of \(\Delta\) satisfies \(\lambda_ 1\geq m\alpha (1-c)/(m-1+c).\) Remark: If \(\rho\) (x)\(\geq \alpha\) for all \(x\in M\), then \(c=0\) so (2)\(\Rightarrow (1)\). Theorem 2 is a generalization of Lichnerowicz's result and gives a lower bound on the lowest eigenvalue if \(\rho\) (x)\(\geq \alpha\) on a large fraction of the manifold.