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 $0\le a\le \frac{1}{2}$ and $\sqrt{1-a^2}\le b\le 1.76$.