Sharp Fundamental Gap Estimate on Convex Domains of Sphere

Fundamental gap estimate for convex domains on sphere -- the case $n=2$

“…This was proven by B. Andrews and J. Clutterbuck in their celebrated work [AC11]. When M = S n , [SWW16] proved the same lower bound for dimensions n ≥ 3 and diameter D < π 2 . The diameter restriction was removed by C. He and the third author in [HW17] by using parabolic methods and a delicate construction of supersolutions to a one-dimensional nonlinear parabolic model.…”

“…The diameter restriction was removed by C. He and the third author in [HW17] by using parabolic methods and a delicate construction of supersolutions to a one-dimensional nonlinear parabolic model. In fact, in the work of [SWW16], the estimate holds for M n K , the simply connected spaces with constant curvature K, with K ≥ 0. In this paper, by using a different model, we show that the fundamental gap estimate for convex domain in S n also holds for n = 2.…”

“…We also give an asymptotic expansion of the first and second Dirichlet eigenvalues of the sphere model in [SWW16]. Recall λ1 (n, D, K), λ2 (n, D, K) are the first and second Dirichlet eigenvalues of…”

“…In §4, we apply the log-concavity result to compare the gap of the first and second eigenvalues between convex domains of spheres and the onedimensional model. In §5, we compute the asymptotics of the first and second eigenvalues of the one-dimensional model used in [SWW16]. The analysis of the one-dimensional model in §5 is interesting on its own and can be read independently.…”

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In [SWW16,HW17] it is shown that the difference of the first two eigenvalues of the Laplacian with Dirichlet boundary condition on convex domain with diameter D of sphere S n is ≥ 3 π 2 D 2 when n ≥ 3. We prove the same result when n = 2.

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