Modulus of Concavity and Fundamental Gap Estimates on Surfaces

2023/06/09 by Khan, Gabriel, Tuerkoen, Malik, Wei, Guofang
#35P15 #53C21 #58J50 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2306.06053

Abstract

The fundamental gap of a domain is the difference between the first two eigenvalues of the Laplace operator. In a series of recent and celebrated works, it was shown that for convex domains in \mathbb Rⁿ and \mathbb Sⁿ with Dirichlet boundary condition the fundamental gap is at least (3 π²)/(D²) where D denotes the diameter of the domain. The key to these results is to establish a strong concavity estimate for the logarithm of the first eigenfunction. In this article, we prove corresponding log-concavity and fundamental gap estimates for surfaces with non-constant positive curvature via a two-point maximum principle. However, the curvature not being constant greatly increases the difficulty of the computation.

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