## Journal of Differential Geometry

### Sharp fundamental gap estimate on convex domains of sphere

#### 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
First available in Project Euclid: 6 June 2019

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