Let the Paneitz operator P0 be strictly positive on a closed 3-manifold M with a fixed conformal class. It is proved that the solution of a fourth-order Q-curvature flow exists on M for all time and converges smoothly to a metric of constant Q-curvature.

In this paper, based on a Harnack-type estimate and a local Sobolev constant bounded for the Calabi flow on closed surfaces, we extend author’s previous results and show the long-time existence and convergence of solutions of 2-dimensional Calabi flow on closed surfaces. Then we establish the uniformization theorem for closed surfaces.

In this paper, firstly, we show the Bondi-mass type estimate of solutions of Calabi flow on closed 4-manifolds. Secondly, in our applications, we obtain the long time existence on closed 4-manifolds. In particular, we are able to show the asymptotic convergence of a subsequence of solutions of the Calabi flow on closed Einstein 4-manifolds.

We construct unbounded positive ${{C}^{2}}$-solutions of the equation $\Delta u\,+\,K{{u}^{\left( n+2 \right)/\left( n-2 \right)}}\,=\,0$ in ${{\mathbb{R}}^{n}}$ (equipped with Euclidean metric ${{g}_{0}}$) such that $K$ is bounded between two positive numbers in ${{\mathbb{R}}^{n}}$, the conformal metric $g\,=\,{{u}^{4/\left( n-2 \right)}}{{g}_{0}}$ is complete, and the volume growth of $g$ can be arbitrarily fast or reasonably slow according to the constructions. By imposing natural conditions on $u$, we obtain growth estimate on the ${{L}^{2n/\left( n-2 \right)}}$-norm of the solution and show that it has slow decay.