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Sharp fundamental gap estimate on convex domains of sphere. (English) Zbl 1418.35286

Summary: In their celebrated work, B. Andrews and J. Clutterbuck proved the fundamental gap (the difference between the first two eigenvalues) conjecture for convex domains in the Euclidean space [Anal. PDE 6, No. 5, 1013–1024 (2013; Zbl 1282.35099)] and conjectured similar results hold for spaces with constant sectional curvature. We prove the conjecture for the sphere. Namely when \(D\), the diameter of a convex domain in the unit \(\mathbb{S}^n\) sphere, is \(\leq \frac{\pi}{2}\), the gap is greater than the gap of the corresponding 1-dim sphere model. We also prove the gap is \(\geq 3 \frac{\pi^2}{D^2}\) when \(n \geq 3\), giving a sharp bound. As in [loc. cit.], the key is to prove a super log-concavity of the first eigenfunction.

MSC:

35P15 Estimates of eigenvalues in context of PDEs
35J25 Boundary value problems for second-order elliptic equations
58J05 Elliptic equations on manifolds, general theory

Citations:

Zbl 1282.35099
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