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

Abstract

We prove that for any complete three-manifold with a lower Ricci curvature bound and a lower bound on the volume of balls of radius one, a solution to the Ricci flow exists for short time. Actually our proof also yields a (non-canonical) way to flow and regularize some interior region of a non-complete initial data satisfying the aformentioned bounds.

Preservation of product structures under the Ricci flow with instantaneous curvature bounds

“…One class of particular interest is that of solutions satisfying a curvature bound of the form c/t for some constant c, which arise naturally as limits of exhaustions (see, e.g., [1], [6], [13]). The purpose of this note is to prove the following.…”

Kähler tori with almost non-negative scalar curvature

“…If the scalar curvature is strengthen to be Ricci curvature, the stability in term of Gromov-Hausdorff topology follows from the celebrated work of Colding [8] using volume the non-collapsing condition. In fact when n = 3, using the Ricci flow method of Hochard [13] and Simon-Topping [21], a sequence of volume non-collapsed metrics g i on T 3 with Ricci curvature bounded from below and almost non-negative scalar curvature will converge to the flat torus in Gromov-Hausdorff topology. On the other hand, if one strengthens the noncollapsing so that g i → g ∞ in C 0 , it was proved by Gromov [10] and Bamler [4] that…”

Existence, lifespan, and transfer rate of Ricci flows on manifolds with small Ricci curvature

“…Existence of Ricci flow solutions on noncompact manifolds was first proved by Shi in [14], where the curvature is assumed to be uniformly bounded. Under weaker curvature conditions and some noncollapsing assumptions, short-time existence has been proved by Xu [19], Simon [16] and Hochard [9], where the curvature may be unbounded initially, but becomes uniformly bounded instantaneously for any positive time. See also the existence result of Chau, Li and Tam in [2] for the Kähler Ricci flow.…”