Explicit fundamental gap estimates for some convex domains in H²

Motivated by an example of Shih [10], we compute the fundamental gap of a family of convex domains in the hyperbolic plane H², showing that for some of them (Formula Presented), where D is the diameter of the domain and λ₁, λ₂ are the first and second Dirichlet eigenvalues of the Laplace operator on the domain. The result contrasts with what is known in Rⁿ or Sⁿ, where (Formula Presented) for convex domains [1, 5, 7, 9]. We also show that the fundamental gap of the example in Shih’s article is still greater than (Formula Presented) , even though the first eigenfunction of the Laplace operator is not log-concave.