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

2017/05/14 by Xian-Tao Huang, Huang, Xian-Tao
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Metric Geometry (math.MG) #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1705.05008

openalex publication_date 2017/05/14 · openalex created_date 2017/05/26 · openalex updated_date 2026/07/28

Abstract

Suppose (Mⁿ,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). We study the behavior of h_d(M) when d is large in this paper. More precisely, suppose that (Mⁿ,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 the asymptotic volume ratio α=lim_R→∞\fracVol(B_p(R))Rⁿ. When α=ω_n, i.e., (Mⁿ,g) is isometric to the Euclidean space, the asymptotic behavior obtained in this paper recovers a well-known asymptotic property of h_d(ℝⁿ).

Citations

• Harmonic functions on metric measure spaces
• Ricci curvature for metric-measure spaces via optimal transport
• On the equivalence of the entropic curvature-dimension condition and Bochner’s inequality on metric measure spaces
• Metric measure spaces with Riemannian Ricci curvature bounded from below
• Lower Bounds on Ricci Curvature and the Almost Rigidity of Warped Products
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• Sharp Hölder continuity of tangent cones for spaces with a lower Ricci curvature bound and applications
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Related

• Harmonic functions with polynomial growth on manifolds with nonnegative Ricci curvature
• Harmonic Functions on Manifolds with Nonnegative Ricci Curvature and Linear Volume Growth
• Volume growth and asymptotic cones of manifolds with nonnegative Ricci curvature
• Sharp Dimension Estimates of Holomorphic Functions and Rigidity
• Harmonic functions on minimal submanifold with cylindrical cone