1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

8. KAHLER-RICCI FLOW WITH NONNEGATIVE BISECTIONAL CURVATURE 103

space of L7:_ 1 is simply en blown up at the origin.) As JzJ ---+ oo, the soliton

metric is asymptotic to a Kahler cone (Cn\{0},§ 8 )/Zk, where e = e(n,k).

For each dimension n :2: 2, Cao [47] adds an n-twisted ]pin-l at JzJ = 0

and constructs a unique complete steady Kahler-Ricci soliton on the total

space of the bundle C '---+ L"!:_n ___,. pn-l. The metric exhibits cigar-paraboloid

behavior at infinity.

For each dimension n :2: 2 and k = n + 1, ... , the authors of [142]

add a pn-l at JzJ = 0 and construct a 1-parameter family of complete

expanding Kahler-Ricci solitons on C '---+ L"!:_k ___,. pn-^1. The solutions are

parameterized bye > 0, where (Cn{O},§e)/Zk is the asymptotic Kahler

cone at infinity.

8. Kahler-Ricci flow with nonnegative bisectional curvature

The study of the Kahler-Ricci fl.ow of Kahler metrics with nonnegative

bisectional curvature is somewhat analogous to the study of the Riemannian

Ricci fl.ow of metrics with nonnegative curvature operator. One aim is to

uniformize Kahler metrics with nonnegative bisectional curvature in both

the compact and noncompact setting. In particular, one would like to fl.ow

such metrics to canonical metrics, or to infer the existence of canonical met-

rics from the long-time behavior of the fl.ow. One would also like to deduce

properties of the underlying complex structure of the Kahler manifold, and

when possible, classify the manifold up to biholomorphism.

8.1. Nonnegative bisectional curvature is preserved. Consider

the Kahler-Ricci fl.ow %t9a;B = -Rap· We shall prove that the Kahler-Ricci

fl.ow preserves the nonnegativity of the bisectional curvature. As in the real

case, the key is Hamilton's weak maximum principle for tensors. This result

was proved first by Banda for n::; 3 and by Mok in any dimension. (See The-

orem 2.69 above.) The result was also extended to the complete noncompact
case by W.-X. Shi under the additional assumption of the bisectional curva-
ture being bounded. We say that a Kahler metric has quasi-positive Ricci

curvature if the Ricci curvature is nonnegative everywhere and positive at

some point.

THEOREM 2.78 (Nonnegative bisectional curvature preserved). The non-

negativity of the bisectional curvature is preserved under the Kahler-Ricci

flow on closed Kahler manifolds. Moreover, if the initial metric also has

quasi-positive Ricci curvature, then both the Ricci curvature and the holo-

morphic sectional curvature are positive for metrics at positive time.

The basic computation in the proof of the above result is the following.