Fundamental gaps of spherical triangles

We compute Dirichlet eigenvalues and eigenfunctions explicitly for spherical lunes and the spherical triangles which are half the lunes, and show that the fundamental gap goes to infinity when the angle of the lune goes to zero. Then we show the spherical equilateral triangle of diameter $frac{pi}{2}$ is a strict local minimizer of the fundamental gap on the space of the spherical triangles with diameter $frac{pi}{2}$, which partially extends Lu-Rowletts result from the plane to the sphere.

The first nonzero eigenvalue of the $p$-Laplacian on Differential forms

we introduce a generalization of the $p$-Laplace operator to act on differential forms and generalize an estimate of Gallot-Meyer for the first nonzero eigenvalue on closed Riemannian manifolds.

Zhong-Yang type eigenvalue estimate with integral curvature condition

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

First eigenvalue of the $p$-Laplacian on Kahler manifolds

We prove a Lichnerowicz type lower bound for the first nontrivial eigenvalue of the $p$-Laplacian on Kahler manifolds. Parallel to the $p = 2$ case, the first eigenvalue lower bound is improved by using a decomposition of the Hessian on Kahler manifolds with positive Ricci curvature.

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

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 $mathbb S^n$ is $geq 3 frac{pi^2}{D^2}$ when $n geq 3$. 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].

First eigenvalue of the $p$-Laplacian under integral curvature condition

We give various estimates of the first eigenvalue of the $p$-Laplace operator on closed Riemannian manifold with integral curvature conditions.

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.

Asymptotic expansion of the Bergman kernel via perturbation of the Bargmann-Fock model

We give an alternate proof of the existence of the asymptotic expansion of the Bergman kernel associated to the $k$-th tensor powers of a positive line bundle $L$ in a $frac{1}{sqrt{k}}$-neighborhood of the diagonal using elementary methods. We use the observation that after rescaling the Kahler potential $kvarphi$ in a $frac{1}{sqrt{k}}$-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.