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Register a new userSharp Spectral Gap and Li-Yaus Estimate on Alexandrov Spaces
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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.
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In this paper, we establish a Bochner type formula on Alexandrov spaces with Ricci curvature bounded below. Yaus gradient estimate for harmonic functions is also obtained on Alexandrov spaces.
In 1997, J. Jost [27] and F. H. Lin [39], independently proved that every energy minimizing harmonic map from an Alexandrov space with curvature bounded from below to an Alexandrov space with non-positive curvature is locally Holder continuous. In [39], F. H. Lin proposed a challenge problem: Can the Holder continuity be improved to Lipschitz continuity? J. Jost also asked a similar problem about Lipschitz regularity of harmonic maps between singular spaces (see Page 38 in [28]). The main theorem of this paper gives a complete resolution to it.
In this paper, we will study the (linear) geometric analysis on metric measure spaces. We will establish a local Li-Yaus estimate for weak solutions of the heat equation and prove a sharp Yaus gradient gradient for harmonic functions on metric measure spaces, under the Riemannian curvature-dimension condition $RCD^*(K,N).$
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.
In this paper, we will prove the Weyls law for the asymptotic formula of Dirichlet eigenvalues on metric measure spaces with generalized Ricci curvature bounded from below.
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