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By A. V Pogorelov, Leo F. Boron

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39J Let (M n , g) be a complete noncompact Kahler manifold with bounded nonnegative holomorphic bisectional curvature. Suppose M has maximum volume growth. Then the scalar curvature decay quadratically. 3) _1_ ( Vx(r) } Bx(r) ndV< - C (1 + r)2 for all x E M and for all r > O. 2, to prove that a non-flat gradient shrinking KahlerRicci soliton with nonnegative bisectional curvature must have zero asymptotic volume ratio. A gradient shrinking Kahler-Ricci soliton is a Kahler 28 A. -F. TAM metric giJ satisfying for some smooth real-valued function f and for some p > O.

O) (M) can be decomposed orthogonally with respect to g(O) as VI EB •.. 1i and . 1i - 1. Hoo t Ivl6 Moreover, the convergences are uniform over all v E Vi \ {O}. 1 ::; i, j ::; 1 and for nonzero vectors v E Vi and W E Vj where i =1= j, limHoo(v(t),w(t))t = 0 and the convergence is uniform over all such nonzero vectors v, w. (ii) For KAHLER-RICCI FLOW AND YAU'S UNIFORMIZATION CONJECTURE 39 (iii) dimc(Vi) = di for each i. (iv) The theorem basically says that the eigenvalues of the Ricci tensor have limits and the eigenspaces are almost the same.

Sympos. Pure Math. 52 Part 2 (1991), 619--625. ca DEPARTMENT OF MATHEMATICS, THE CHINESE UNIVERSITY OF HONG KONG, SHATIN, HONG KONG, CHINA. hk Surveys in Differential Geometry XII Recent Developments on Hamilton's Ricci flow Huai-Dong Cao, Bing-Long Chen, Xi-Ping Zhu ABSTRACT. In 1982, Hamilton [41] introduced the Ricci flow to study compact three-manifolds with positive Ricci curvature. Through decades of works of many mathematicians, the Ricci flow has been widely used to study the topology, geometry and complex structure of manifolds.

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