scholarly journals Harmonic functions on manifolds with nonnegative Ricci curvature and linear volume growth

2000 ◽  
Vol 192 (1) ◽  
pp. 183-189 ◽  
Author(s):  
Christina Sormani
Keyword(s):  
Ricci Curvature ◽  
Harmonic Functions ◽  
Volume Growth ◽  
Nonnegative Ricci Curvature

Gradient Estimates for Harmonic Functions on Manifolds With Lipschitz Metrics

1998 ◽  
Vol 50 (6) ◽  
pp. 1163-1175 ◽  
Author(s):  
Jingyi Chen ◽  
Elton P. Hsu
Keyword(s):  
Ricci Curvature ◽  
Harmonic Functions ◽  
Volume Growth ◽  
Gradient Estimate ◽  
Gradient Estimates ◽  
Smooth Manifolds ◽  
Lipschitz Continuous ◽  
Bounded Below

AbstractWe introduce a distributional Ricci curvature on complete smooth manifolds with Lipschitz continuous metrics. Under an assumption on the volume growth of geodesics balls, we obtain a gradient estimate for weakly harmonic functions if the distributional Ricci curvature is bounded below.

scholarly journals On the asymptotic behavior of the dimension of spaces of harmonic functions with polynomial growth

2020 ◽  
Vol 2020 (762) ◽  
pp. 281-306 ◽  
Author(s):  
Xian-Tao Huang
Keyword(s):  
Asymptotic Behavior ◽  
Harmonic Functions ◽  
Tangent Cone ◽  
Polynomial Growth ◽  
Volume Growth ◽  
Volume Ratio ◽  
Growth Order ◽  
Maximal Volume ◽  
Nonnegative Ricci Curvature ◽  
Asymptotic Volume

AbstractSuppose {(M^{n},g)} is a Riemannian manifold with nonnegative Ricci curvature, and let {h_{d}(M)} be the dimension of the space of harmonic functions with polynomial growth of growth order at most d. Colding and Minicozzi proved that {h_{d}(M)} is finite. Later on, there are many researches which give better estimates of {h_{d}(M)}. In this paper, we study the behavior of {h_{d}(M)} when d is large. More precisely, suppose {(M^{n},g)} has maximal volume growth and has a unique tangent cone at infinity. Then when d is sufficiently large, we obtain some estimates of {h_{d}(M)} in terms of the growth order d, the dimension n and the asymptotic volume ratio {\alpha=\lim_{R\rightarrow\infty}\frac{\mathrm{Vol}(B_{p}(R))}{R^{n}}}. When {\alpha=\omega_{n}}, i.e., {(M^{n},g)} is isometric to the Euclidean space, the asymptotic behavior obtained in this paper recovers a well-known asymptotic property of {h_{d}(\mathbb{R}^{n})}.

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