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

Abstract

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.

Citation

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Shoo Seto. Lili Wang. Guofang Wei. "Sharp fundamental gap estimate on convex domains of sphere." J. Differential Geom. 112 (2) 347 - 389, June 2019. https://doi.org/10.4310/jdg/1559786428

Information

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

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Vol.112 • No. 2 • June 2019
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