THE FUNDAMENTAL GAP OF SIMPLICES
Zhiqin Lu, Julie Rowlett · 2013
Abstract. The gap function of a domain Ω ⊂ Rn is ξ(Ω): = d2(λ2 − λ1), where d is the diameter of Ω, and λ1 and λ2 are the first two positive Dirichlet eigenvalues of the Euclidean Laplacian on Ω. It was recently shown by Andrews and Clutterbuck [1] that for any convex Ω ⊂ Rn, ξ(Ω) ≥ 3pi2, where the infimum occurs for n = 1. On the other hand, the gap function on the moduli space of n-simplices behaves differently. Our first theorem is a compactness result for the gap function on the moduli space of n-simplices. Next, specializing to n = 2, our second main result proves the recent conjecture of Antunes-Freitas [2]: for any triangle T ⊂ R2, ξ(T) ≥ 64pi