Open Access
June 2019 Sharp fundamental gap estimate on convex domains of sphere
Shoo Seto, Lili Wang, Guofang Wei
Author Affiliations +
J. Differential Geom. 112(2): 347-389 (June 2019). DOI: 10.4310/jdg/1559786428


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 [3] 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 [3], the key is to prove a super log-concavity of the first eigenfunction.

Funding Statement

S. Shoo was partially supported by a Simons Travel Grant.
L. Wang was partially supported by CNSF Grant 11671141 and by the Geometry Center at ECNU.
G. Wei was partially supported by NSF DMS 1506393.


Download Citation

Shoo Seto. Lili Wang. Guofang Wei. "Sharp fundamental gap estimate on convex domains of sphere." J. Differential Geom. 112 (2) 347 - 389, June 2019.


Received: 19 June 2016; Published: June 2019
First available in Project Euclid: 6 June 2019

zbMATH: 07064406
MathSciNet: MR3960269
Digital Object Identifier: 10.4310/jdg/1559786428

Rights: Copyright © 2019 Lehigh University

Vol.112 • No. 2 • June 2019
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