An improved upper bound for the second eigenvalue on tori

2025/06/06 by Kang, Fan
#Differential Geometry (math.DG) #FOS: Mathematics #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2506.05846

Abstract

In this paper, we consider the problem of maximizing the second non-zero eigenvalue λ₂(T,g) of the Laplace-Beltrami operator on a torus (T,g), among all unit-area metrics in a fixed conformal class. Based on work by Karpukhin-Stern and Eddaoudi-Girouard, we give an explicit upper bound for λ₂(T,g) in a fixed conformal class of the torus. Our bound improves previous estimates based on conformal area and shows that, in the majority of conformal classes, the second eigenvalue is strictly less than (8π²)/(√(3))+8π, supporting the conjecture of Kao-Lai-Osting that this value is the supremum of λ₂(T,g) over all conformal classes. Moreover, we give the uniform upper bound λ₂(T,g)< (16π²)/(√(3)) for all unit-area metrics g on T.

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