Harmonic functions with polynomial growth on manifolds with nonnegative Ricci curvature

2021/09/15 by Xian-Tao Huang, Huang, Xian-Tao · 1 citation
Mathematics · #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.2109.07534

Abstract

Suppose (M,g) is a Riemannian manifold having dimension n, nonnegative Ricci curvature, maximal volume growth and unique tangent cone at infinity. In this case, the tangent cone at infinity C(X) is an Euclidean cone over the cross-section X. Denote by α=lim_r→∞\fracVol(B_r(p))rⁿ the asymptotic volume ratio. Let h_k=h_k(M) be the dimension of the space of harmonic functions with polynomial growth of growth order at most k. In this paper, we prove a upper bound of h_k in terms of the counting function of eigenvalues of X. As a corollary, we obtain lim_k→∞k^1-nh_k=\frac2α(n-1)!ω_n. These results are sharp, as they recover the corresponding well-known properties of h_k(ℝⁿ). In particular, these results hold on manifolds with nonnegative sectional curvature and maximal volume growth.

Citations

• On the equivalence of the entropic curvature-dimension condition and Bochner’s inequality on metric measure spaces
• Differential equations on riemannian manifolds and their geometric applications
• Lower Bounds on Ricci Curvature and the Almost Rigidity of Warped Products
• The splitting theorem in non-smooth context

Cited by

• Harmonic functions on minimal submanifold with cylindrical cone

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