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 S n sphere, is ≤ π 2 , the gap is greater than the gap of the corresponding 1-dim sphere model. We also prove the gap is ≥ 3 π 2 D 2 when n ≥ 3, giving a sharp bound. As in [3], the key is to prove a super log-concavity of the first eigenfunction. π 2 D 2 , see also [21]. Only in 2011, the conjecture was completely solved by B. Andrews and J. Clutterbuck in their celebrated work [3] by establishing a sharp log-concavity estimate for the first eigenfunction, see also [23]. For convex domains on a sphere, Lee and Wang [20] showed the gap is ≥ π 2 D 2 . See [24] for an estimate on general manifolds.Key words and phrases. Eigenvalue estimate, spectral gap. Partially supported by NSF DMS 1506393.the eigenvalue and eigenfunction of the 1-dimensional model can not be solved explicitly. We obtained a gap estimate for the model by obtaining several monotonicity properties for the eigenvalues and eigenfunctions. In §3, we prove Theorem 1.5, the key super log-concavity estimate. Following [3], the idea is to apply the maximum principle to the so called two-point functions. For K = 0, the computation is much more subtle. In R n , Andrews-Clutterbuck proved the preservation of modulus holds for general solutions of the heat equation. It is not clear if this is true when K = 0. We use both the heat equation and the Laplacian equation to prove several preservation of modulus. Several elliptic versions are also obtained.Finally in §4, with the log-concavity result we derive a gap comparison for general manifolds with lower Ricci curvature bound. Namely the gap of the Laplacian is greater or equal to the gap of the 1-dimensional model, thereby proving Theorem 1.1. We give two proofs of the gap comparison, one elliptic and one parabolic. As another application of (1.4) we also give a lower bound on the first Dirichlet eigenvalue of the Laplacian on convex domain in sphere, see Proposition 4.6.

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Abstract. We give an alternate proof of the existence of the asymptotic expansion of the Bergman kernel associated to the kth tensor powers of a positive line bundle L in a 1 √ kneighborhood of the diagonal using elementary methods. We use the observation that after rescaling the Kähler potential kϕ in a 1 √ k -neighborhood of a given point, the potential becomes an asymptotic perturbation of the Bargmann-Fock metric. We then prove that the Bergman kernel is also an asymptotic perturbation of the Bargmann-Fock Bergman kernel.

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. In fact our proof works for all dimension. We also give an asymptotic expansion of the first and second Dirichlet eigenvalues of the model in [SWW16].

We prove a sharp Zhong-Yang type eigenvalue lower bound for closed Riemannian manifolds with control on integral Ricci curvature.

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