Closed n-manifolds with diam² sec ≥ -κ and diam² Ric ≥ -δ (δ small depending on n,κ) fiber over a b1(M)-torus, removing the upper sectional curvature bound from Yamaguchi's prior result.
Harmonic functions on metric measure spaces
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abstract
In this paper, we study harmonic functions on metric measure spaces with Riemannian Ricci curvature bounded from below, which were introduced by Ambrosio-Gigli-Savar\'e. We prove a Cheng-Yau type local gradient estimate for harmonic functions on these spaces. Furthermore, we derive various optimal dimension estimates for spaces of polynomial growth harmonic functions on metric measure spaces with nonnegative Riemannian Ricci curvature.
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Proves Lipschitz regularity of continuous harmonic maps from finite-dimensional Alexandrov spaces to compact smooth Riemannian manifolds, solving Lin's conjecture by extending Huang-Wang's argument.
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Partial regularity of harmonic maps from Alexandrov spaces
Proves Lipschitz regularity of continuous harmonic maps from finite-dimensional Alexandrov spaces to compact smooth Riemannian manifolds, solving Lin's conjecture by extending Huang-Wang's argument.