96 2. KAHLER-RICCI FLOW
THEOREM 2. 70 (KRF: compact positive bisectional curvature). Suppose
(Mn, g ( 0)) is a closed Kahler manifold with nonnegative bisectional curva-
ture everywhere and positive bisectional curvature at a point. Then the solu-
tion g (t) to the normalized Kahler-Ricci flow, which has positive bisectional
curvature for all t > 0, converges exponentially fast to the Fubini-Study
metric of constant holomorphic sectional curvature on ccpn.
REMARK 2.71. Without using the existence of a Kahler-Einstein metric,
Cao, Chen, and Zhu [49] proved a uniform curvature estimate (see Theorem
2.92).
6.2. Uniformization of noncompact Kahler manifolds with non-
negative bisectional curvature. In this subsection we recall Yau's fun-
damental conjecture on the uniformization of complete noncompact Kahler
manifolds with nonnegative bisectional curvature.
CONJECTURE 2.72 (Noncompact Kahler uniformization Kc (V, W) > 0).
If (Mn, g (0)) is a complete noncompact Kahler manifold with positive bi-
sectional curvature, then M is biholomorphic to ccn.
Using the Kahler-Ricci flow on noncompact manifolds, Chau and Tam
[65] proved the following result, which affirms Yau's conjecture in the case
of bounded curvature and maximum volume growth.
THEOREM 2. 73 (KRF: noncompact positive bisectional curvature). If
(Mn, g (0)) is a complete noncompact Kahler manifold with bounded positive
bisectional curvature and maximum volume growth, then M is biholomorphic
to en.
There have been a number of works on the Kahler-Ricci flow on non-
compact manifolds with positive bisectional curvature. For example, the
reader may consult Shi [331], Tam and one of the authors [290], [292], and
Chen and Zhu [79], [80].
6.3. Limiting behavior of the Kahler-Ricci fl.ow on closed man-
ifolds. There are also the following results about the limiting behavior of
the Kahler-Ricci flow due to Sesum [323].
THEOREM 2.74. If(Mn,g(t)), t E [O,oo), is a solution to the Kahler-
Ricci flow on a closed manifold with uniformly bounded Ricci curvature, then
for any sequence ti~ oo there exists a subsequence such that (M, g (t +ti))
converges to (M~, g 00 (t)), where g 00 (t) is a solution to the Kahler-Ricci
flow. The convergence is outside a set of real codimension 4.
When n = 2, Sesum improved the above result to the following.THEOREM 2.75. If (M^2 ,g(t)), t E [O,oo), is a solution to the Kahler-
Ricci flow on a closed manifold with uniformly bounded Ricci curvature, then
for any sequence ti ~ oo there exists a subsequence such that (M, g (t +ti))
converges to (M~, g 00 (t)) , where g 00 (t) is a Kahler-Ricci soliton. The
convergence is outside a finite number of points.