scholarly journals Ricci flow and the uniformization on complete noncompact Kähler manifolds

1997 ◽  
Vol 45 (1) ◽  
pp. 94-220 ◽  
Author(s):  
Wan-Xiong Shi
Keyword(s):  
Ricci Flow ◽  
Kähler Manifolds ◽  
Kahler Manifolds

scholarly journals POSITIVITY OF RICCI CURVATURE UNDER THE KÄHLER–RICCI FLOW

2006 ◽  
Vol 08 (01) ◽  
pp. 123-133 ◽  
Author(s):  
DAN KNOPF
Keyword(s):  
Ricci Curvature ◽  
Ricci Flow ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Bounded Curvature ◽  
Complex Dimension

In each complex dimension n ≥ 2, we construct complete Kähler manifolds of bounded curvature and non-negative Ricci curvature whose Kähler–Ricci evolutions immediately acquire Ricci curvature of mixed sign.

scholarly journals A C0-estimate for the parabolic Monge–Ampère equation on complete non-compact Kähler manifolds

2009 ◽  
Vol 146 (1) ◽  
pp. 259-270 ◽  
Author(s):  
Albert Chau ◽  
Luen-Fai Tam
Keyword(s):  
Ricci Flow ◽  
Kähler Manifolds ◽  
Main Step ◽  
Kahler Manifolds ◽  
Compact Sets ◽  
Ricci Flat ◽  
Long Time ◽  
Flat Manifolds ◽  
Monge Ampere ◽  
Invent Math

AbstractIn this article we study the Kähler–Ricci flow, the corresponding parabolic Monge–Ampère equation and complete non-compact Kähler–Ricci flat manifolds. Our main result states that if (M,g) is sufficiently close to being Kähler–Ricci flat in a suitable sense, then the Kähler–Ricci flow has a long time smooth solution g(t) converging smoothly uniformly on compact sets to a complete Kähler–Ricci flat metric on M. The main step is to obtain a uniform C0-estimate for the corresponding parabolic Monge–Ampère equation. Our results on this can be viewed as parabolic versions of the main results of Tian and Yau [Complete Kähler manifolds with zero Ricci curvature. II, Invent. Math. 106 (1990), 27–60] on the elliptic Monge–Ampère equation.

Holomorphic Functions on a Class of Kähler Manifolds With Nonnegative Curvature, I

2020 ◽  
Author(s):  
Bo Yang
Keyword(s):  
Asymptotic Behavior ◽  
Ricci Curvature ◽  
Ricci Flow ◽  
Polynomial Growth ◽  
Holomorphic Functions ◽  
Nonnegative Curvature ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Nonnegative Ricci Curvature

Abstract In this paper, we consider holomorphic functions of polynomial growth on complete Kähler manifolds with nonnegative curvature. We explain how their growth orders are related to the asymptotic behavior of Kähler–Ricci flow. The main result is to determine minimal orders of holomorphic functions on gradient Kähler–Ricci expanding solitons with nonnegative Ricci curvature.

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