Communications in Analysis and Geometry

Volume 12 (2004)

Number 1

Kähler-Ricci Flow and the Poincaré-Lelong Equation

Pages: 111 – 141

DOI: https://dx.doi.org/10.4310/CAG.2004.v12.n1.a7

Authors

Lei Ni

Luen-Fai Tam

Abstract

In [M-S-Y], Mok-Siu-Yau studied complete Kähler manifolds with nonnegative holomorphic bisectional curvature by solving the Poincaré-Lelong equation

√ - 1δ \overscore δu = Ric (0.1)

where Ric is the Ricci form of the manifold. In [M-S-Y], the authors solved (0.1) under the assumptions that the manifold is of maximal volume growth and the scalar curvature decays quadratically. On the other hand, in a series of papers of W.-X. Shi [Sh2-4], Kähler-Ricci flow

{δ \over δt} γα\overscore β = - Rα\overscore β (0.2)

has been studied extensively and important applications were given. In [N1] and [N-S-T], the Poincaré-Lelong equation has been solved under more general conditions than in [M-S-Y]. The conditions in [N-S-T] are more in line with the conditions in [Sh2-4]. Since a solution of (0.1) is a potential for the Ricci tensor, it is interesting to see if one can apply (0.1) to study solutions of (0.2).

In this work, on the one hand we shall study the Kähler-Ricci flows by using solutions of the Poincaré-Lelong equation.

Published 1 January 2004