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Sharp Fundamental Gap Estimate on Convex Domains of Sphere

Authors :
Seto, Shoo
Wang, Lili
Wei, Guofang
Publication Year :
2016

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 and conjectured similar results holds 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 $S^n$ sphere, is $\le \frac{\pi}{2}$, the gap is greater than the gap of the corresponding $1$-dim sphere model. We also prove the gap is $\ge 3\frac{\pi^2}{D^2}$ when $n \ge 3$, giving a sharp bound. As in Andrews-Clutterbuck's proof of the fundamental gap, the key is to prove a super log-concavity of the first eigenfunction.<br />Comment: An error and typos corrected, modified the proof of Theorem 1.5, Theorem 3.2, and Appendix B added

Details

Database :
arXiv
Publication Type :
Report
Accession number :
edsarx.1606.01212
Document Type :
Working Paper