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Register a new userSharp Fundamental Gap Estimate on Convex Domains of Sphere
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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.
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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 $mathbb S^n$ 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].
In the previous work [35], the second and third authors established a Bochner type formula on Alexandrov spaces. The purpose of this paper is to give some applications of the Bochner type formula. Firstly, we extend the sharp lower bound estimates of spectral gap, due to Chen-Wang [9, 10] and Bakry-Qian [6], from smooth Riemannian manifolds to Alexandrov spaces. As an application, we get an Obata type theorem for Alexandrov spaces. Secondly, we obtain (sharp) Li-Yaus estimate for positve solutions of heat equations on Alexandrov spaces.
Adapting the method of Andrews-Clutterbuck we prove an eigenvalue gap theorem for a class of non symmetric second order linear elliptic operators on a convex domain in euclidean space. The class of operators includes the Bakry-Emery laplacian with potential and any operator with second order term the laplacian whose first order terms have coefficients with compact support in the open domain. The eigenvalue gap is bounded below by the gap of an associated Sturm-Liouville problem on a closed interval.
We prove that the static convexity is preserved along two kinds of locally constrained curvature flows in hyperbolic space. Using the static convexity of the flow hypersurfaces, we prove new family of geometric inequalities for such hypersurfaces in hyperbolic space.
In this note, we prove the sharp Davies-Gaffney-Grigoryan lemma for minimal heat kernels on graphs.
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