We consider the Ricci flow $frac{partial}{partial t}g=-2Ric$ on the 3-dimensional complete noncompact manifold $(M,g(0))$ with non-negative curvature operator, i.e., $Rmgeq 0, |Rm(p)|to 0, ~as ~d(o,p)to 0.$ We prove that the Ricci flow on such a manifold is nonsingular in any finite time.
In this paper, we employ a nonlocal $Q$-curvature flow inspired by Gursky-Malchiodis work cite{gur_mal} to solve the prescribed $Q$-curvature problem on a class of closed manifolds: For $n geq 5$, let $(M^n,g_0)$ be a smooth closed manifold, which is
not conformally diffeomorphic to the standard sphere, satisfying either Gursky-Malchiodis semipositivity hypotheses: scalar curvature $R_{g_0}>0$ and $Q_{g_0} geq 0$ not identically zero or Hang-Yangs: Yamabe constant $Y(g_0)>0$, Paneitz-Sobolev constant $q(g_0)>0$ and $Q_{g_0} geq 0$ not identically zero. Let $f$ be a smooth positive function on $M^n$ and $x_0$ be some maximum point of $f$. Suppose either (a) $n=5,6,7$ or $(M^n,g_0)$ is locally conformally flat; or (b) $n geq 8$, Weyl tensor at $x_0$ is nonzero. In addition, assume all partial derivatives of $f$ vanish at $x_0$ up to order $n-4$, then there exists a conformal metric $g$ of $g_0$ with its $Q$-curvature $Q_g$ equal to $f$. This result generalizes Escobar-Schoens work [Invent. Math. 1986] on prescribed scalar curvature problem on any locally conformally flat manifolds of positive scalar curvature.
We show some results for the $L^2$ curvature flow linked by the theme of addressing collapsing phenomena. First we show long time existence and convergence of the flow for $SO(3)$-invariant initial data on $S^3$, as well as a long time existence and
convergence statement for three-manifolds with initial $L^2$ norm of curvature chosen small with respect only to the diameter and volume, which are both necessary dependencies for a result of this kind. In the critical dimension $n = 4$ we show a related low-energy convergence statement with an additional hypothesis. Finally we exhibit some finite time singularities in dimension $n geq 5$, and show examples of finite time singularities in dimension $n geq 6$ which are collapsed on the scale of curvature.