Explicit fundamental gap estimates for some convex domains in $\mathbb H^2$
Theodora Bourni, Julie Clutterbuck, Xuan Hien Nguyen, Alina Stancu, Guofang Wei, Valentina‐Mira Wheeler · arXiv (Cornell University) · 2019
Motivated by an example of Shih, we compute the fundamental gap of a family of convex domains in the hyperbolic plane $\mathbb H^2$, showing that for some of them $λ_2 - λ_1 < \frac{3π^2}{D^2}$, where $D$ is the diameter of the domain and $λ_1$, $λ_2$ are the first and second Dirichlet eigenvalues of the Laplace operator on the domain. The result contrasts with what is known in $\mathbb R^n $ or $\mathbb S^n$, where $λ_2 - λ_1 \geq \frac{3 π^2}{D^2}$ for convex domains. We also show that the fundamental gap of the example in Shih's article is still greater than $\tfrac 32 \frac{π^2}{D^2}$, even though the first eigenfunction of the Laplace operator is not log-concave.