BOUNDING SCALAR CURVATURE AND DIAMETER ALONG THE KÄHLER RICCI FLOW (AFTER PERELMAN)
In this short note we present a result of Perelman with detailed proof. The result states that if $g(t)$ is the Kähler Ricci flow on a compact, Kähler manifold $M$ with $c_1(M)>0$, the scalar curvature and diameter of $(M,g(t))$ stay uniformly bounded along the flow, for $t\in[0,\infty)$. We learned about this result and its proof from Grigori Perelman when he was visiting MIT in the spring of 2003. This may be helpful to people studying the Kähler Ricci flow.(Published Online April 10 2008)
(Received November 22 2006)
(Revised December 16 2007)
(Accepted January 4 2008)
Key Words: scalar curvature; diameter; Kähler Ricci flow.