Analysis Of Solutions To The Ricci Flow Which Are Initially Singular

In our paper, one of the topics of study is the initial blow-up problem of n-dimensional solutions of the Ricci flow with nonnegative curvature operator, i.e., the solution of the Ricci flow blows up at time t = 0 in the sense that the curvature tends to infinity as t↘ 0.Main Theorem: Let (M~n, g(t)) be a complete solution to the unnormalized Ricci flow which defined (0, T] with nonnegative curvature operator. Then we can find sequence (x_i,t_i,) and t_i→0 , the sequence (M~2,g_i(t),x_i) preconverges to a complete immortal solution (M_∞~2,g_∞(t),x_∞) , with bounded curvature. The singularity model (M_∞~2, g_∞(t)) is an expanding Ricci soliton.The main difficulty of our work is to deal with the non-collapsing phenomena of the Ricci flow as time approaches the singular time (t = 0). Inspired by Perel-man's work [17], we have given two proofs, one using the expander entropy and the other using the forward reduced volume based at the singular time t = 0.