Journal of Differential Geometry

Sharp fundamental gap estimate on convex domains of sphere

Shoo Seto, Lili Wang, and Guofang Wei

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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.

Note

S. Shoo was partially supported by a Simons Travel Grant.

Note

L. Wang was partially supported by CNSF Grant 11671141 and by the Geometry Center at ECNU.

Note

G. Wei was partially supported by NSF DMS 1506393.

Article information

Source
J. Differential Geom., Volume 112, Number 2 (2019), 347-389.

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

Permanent link to this document
https://projecteuclid.org/euclid.jdg/1559786428

Digital Object Identifier
doi:10.4310/jdg/1559786428

Mathematical Reviews number (MathSciNet)
MR3960269

Zentralblatt MATH identifier
07064406

Citation

Seto, Shoo; Wang, Lili; Wei, Guofang. Sharp fundamental gap estimate on convex domains of sphere. J. Differential Geom. 112 (2019), no. 2, 347--389. doi:10.4310/jdg/1559786428. https://projecteuclid.org/euclid.jdg/1559786428


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