Sharp Fundamental Gap Estimate on Convex Domains of Sphere

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 $ge 3frac{pi^2}{D^2}$ when $n ge 3$, giving a sharp bound. As in Andrews-Clutterbucks proof of the fundamental gap, the key is to prove a super log-concavity of the first eigenfunction.