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.

中文翻译：

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球面凸域的尖锐基本间隙估计

在他们著名的工作中，B. Andrews 和 J. Clutterbuck 证明了欧几里德空间中凸域的基本间隙（前两个特征值之间的差异）猜想，并推测类似的结果适用于具有恒定截面曲率的空间。我们证明球体的猜想。即当单位$S^n$球体内凸域的直径$D$为$\le\frac{\pi}{2}$时，间隙大于对应的$1$-昏暗的球体模型。我们还证明了当 $n \ge 3$ 时的差距是 $\ge 3\frac{\pi^2}{D^2}$，给出了一个尖锐的界限。正如安德鲁斯-克拉特巴克对基本差距的证明一样，关键是要证明第一本征函数的超对数凹度。