An improved upper bound for the second eigenvalue on tori

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 $\sqrt{1-a^2}\leq b\leq 1.76$.