2024/04/12 by Gang Liu, Liu, Gang
Health Professions · Mathematics · #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Therapeutic Uses of Natural Elements
paper · pdf · doi:10.48550/arxiv.2404.08537
openalex publication_date 2024/04/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider complete Kähler manifolds with nonnegative Ricci curvature. The main results are: 1. When the manifold has nonnegative bisectional curvature, we show that limr→∞\fracr2vol(B(p, r))∫B(p, r)S exists. In other words, it depends only on the manifold. This solves a question of Ni. Also, we establish estimates among volume growth ratio, integral of scalar curvature, and the degree of polynomial growth holomorphic functions. The new point is that the estimates are sharp for any prescribed volume growth rate. 2. We discover a strong rigidity for complete Ricci flat Kähler metrics. Let Mn (n≥ 2) be a complete Kähler manifold with nonnegative Ricci curvature and Euclidean volume growth. Assume either the curvature has quadratic decay, or the Kähler metric is ddc-exact with quadratic decay of scalar curvature. If one tangent cone at infinity is Ricci flat, then M is Ricci flat. In particular, the tangent cone is unique. In other words, we can test Ricci flatness of the manifold by checking one single tangent cone. This seems unexpected, since apriori, there is no equation on M and the Bishop-Gromov volume comparison is not sharp on Ricci flat (nonflat) manifolds. Such result is in sharp contrast to the Riemannian setting: Colding and Naber showed that tangent cones are quite flexible when Ric≥ 0 and |Rm|r2