Lower bound of Ricci flow's existence time

Abstract

Let (M n , g) be a compact n-dimensional (n 2) manifold with nonnegative Ricci curvature, and if n 3, then we assume that (M n , g) × R has nonnegative isotropic curvature. The lower bound of the Ricci flow's existence time on (M n , g) is proved. This provides an alternative proof for the uniform lower bound of a family of closed Ricci flows' maximal existence times, which was first proved by E. Cabezas-Rivas and B. Wilking. We also get an interior curvature estimate for n = 3 under Rc 0 assumption among others. Combining these results, we proved the shorttime existence of the Ricci flow on a large class of three-dimensional open manifolds, which admit some suitable exhaustion covering and have nonnegative Ricci curvature.

Short-time existence of the Ricci flow on complete, non-collapsed $3$-manifolds with Ricci curvature bounded from below

“…Like any ancient 3-dimensional solution to the Ricci flow, h(s) has non-negative sectional curvature ([9], 6.50 and [7]), while on the other hand (18) implies that for all r > 0, vol h(0) B 0 (x, r) ≥ vr 3 , that is to say that the Asymptotic Volume Ratio (AVR, see note page 7) of h(0) is strictly positive. According to Perelman ([11], 11.4), the only possibility is that (M, h(s)) is the trivial static solution given by the Euclidian metric (R 3 , g E 3 ).…”

“…A uniform bound on the existence time and additional estimates were proved by X. Guoyi in [18] for the family of those compact Riemannian manifolds satisfying condition (C 3 ), yet the non-compact case remains open, whence the following Question 1.3. Let (M 3 , g 0 , x 0 ) be a complete non-compact manifold satisfying condition (C 3 ).…”