Chern–Ricci flows on noncompact complex manifolds

Abstract

In this work, we obtain existence criteria for Chern–Ricci flows on noncompact manifolds. We generalize a result by Tossati–Wienkove [37] on Chern-Ricci flows to noncompact manifolds and a result for Kähler–Ricci flows by Lott–Zhang [21] to Chern–Ricci flows. Using the existence results, we prove that any complete noncollapsed Kähler metric with nonnegative bisectional curvature on a noncompact complex manifold can be deformed to a complete Kähler metric with nonnegative and bounded bisectional curvature which will have maximal volume growth if the initial metric has maximal volume growth. Combining this result with [3], we give another proof that a complete noncompact Kähler manifold with nonnegative bisectional curvature (not necessarily bounded) and maximal volume growth is biholomorphic to $\mathbb{C}^n$. This last result has already been proved by Liu [20] recently using other methods. This last result is a partial confirmation of a uniformization conjecture of Yau [41].