Fundamental gap estimate for convex domains on sphere -- the case $n=2$

Abstract

In [SWW16,HW17] it is shown that the difference of the first two eigenvalues of the Laplacian with Dirichlet boundary condition on convex domain with diameter D of sphere S n is ≥ 3 π 2 D 2 when n ≥ 3. We prove the same result when n = 2. In fact our proof works for all dimension. We also give an asymptotic expansion of the first and second Dirichlet eigenvalues of the model in [SWW16].

The vanishing of the fundamental gap of convex domains in $\mathbb H^n$

“…We refer to this paper for history and earlier work on this important subject, see also the survey article [7]. More recently, Dai, He, Seto, Wang, and Wei (in various subsets) [6,10,14] generalized the fundamental gap estimate to convex domains in S n , showing that the same bound holds:…”

The fundamental gap of horoconvex domains in $\mathbb H^n$

“…with Dirichlet boundary conditions at 0 and r, where the second eigenvalue λ 2 is the first eigenvalue of (7). Using once more the ODE comparison theorem, we obtain…”

“…In this article, the fundamental gap of a domain is the difference between the first two eigenvalues of the Laplacian with zero Dirichlet boundary conditions. For convex domains in R n or S n , n ≥ 2, it is known from [1,7,9,13] that λ 2 − λ 1 ≥ 3π 2 /D 2 , where D is the diameter of the domain.…”

Explicit fundamental gap estimates for some convex domains in $\mathbb H^2$

“…Recently, Dai, He, Seto, Wang, and Wei (in various subsets) [10,5,8] generalized the fundamental gap estimate to convex domains in S n , showing that…”

“…D 2 , in order to obtain the optimal estimates in [1,10,5,8] it is shown that the first eigenfunction is super log-concave, namely that the first eigenfunction is more logconcave than the first eigenfunction of the following one-dimensional model operator,…”