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

Let $(M^{n}, g)$ be a closed connected Einstein space, $n=dim M ,$ and $\kappa_{0}$ be the lower bound of the sectional curvature. In this paper, we prove Udo Simon's conjecture: on closed Einstein spaces, $n\geq 3,$ 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.