Chern-ricci flows on noncompact complex manifolds

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 non-collapsed 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 Cⁿ. 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].