Curvature bounds for the spectrum of closed Einstein spaces and Simon conjecture

Abstract

Let $(M^n,g)$ be a closed connected Einstein space, $n=dimM,$ and ${\kappa }_0$ be the lower bound of the sectional curvature. In this paper, we prove Udo Simon's conjecture: on closed Einstein spaces, $n\ge 3,$ there is no eigenvalue $\lambda$ such that $$n{\kappa }_0<\lambda <2(n+1){\kappa }_0,$$ and both bounds are the best possible. Furthermore, we develop Simon's conjecture to the next gap of eigenvalue $\lambda :$ on closed Einstein spaces, there is no $\lambda$ such that $$2(n+1){\kappa }_0<\lambda <2(n+2){\kappa }_0,$$ and both bounds are the best possible.

9 pages