An improved upper bound for the second eigenvalue on tori

TL;DR

This paper improves the upper bounds for the second eigenvalue of the Laplacian on tori, providing new estimates that refine previous results and have implications for related conjectures.

Contribution

The authors derive a new upper bound for the second Laplace eigenvalue on flat tori, improving existing estimates and reducing a conjecture to a specific case.

Findings

Established a new upper bound for $oldsymbol{ ext{lambda}_2}$ on flat tori.

Derived a universal upper bound $oldsymbol{ ext{lambda}_2(T,g)< rac{16 ext{pi}^2}{ ext{sqrt}(3)}$.

Reduced the Kao-Lai-Osting conjecture to a specific family of flat tori.

Abstract

In this paper, we study the maximization problem of the second non-zero Laplace eigenvalue $\lambda_2(T,g)$ on a torus $T$, among all unit-area metrics in a fixed conformal class. First, we obtain a new upper bound for $\lambda_2(T_{a,b},g)$ on any flat torus $T_{a, b}$ with $(a, b)\in \mathbb{R}^2$. Our bound improves the general estimate $\lambda_2(T_{a, b},g)\le 4A_c(T_{a, b}, [g])$ in the case of the torus. As applications, we derive a uniform upper bound $\lambda_2(T,g)< \frac{16\pi^2}{\sqrt{3}}$ for any torus $T$ and any metric $g$, and reduce the Kao-Lai-Osting conjecture to proving an upper bound for $\lambda_2(T_{a,b},g)$ on the specific family of flat tori $T_{a,b}$ with $0\leq a\leq \frac12$ and $\sqrt{1-a^2}\leq b\leq 1.76$.