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

TL;DR

This paper proves Simon's conjecture regarding the spectral gap bounds for eigenvalues of closed Einstein spaces, establishing optimal bounds and extending the conjecture to the next eigenvalue gap.

Contribution

It confirms Simon's conjecture on the non-existence of eigenvalues within specific spectral gaps for closed Einstein spaces and extends the result to a larger eigenvalue interval.

Findings

Proves no eigenvalues exist between nκ₀ and 2(n+1)κ₀.

Establishes the bounds are optimal.

Extends the spectral gap result to the interval 2(n+1)κ₀ and 2(n+2)κ₀.

Abstract

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.