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Chern-Ricci flows on noncompact complex manifolds. (English) Zbl 1469.32017

A famous conjecture of S.-T. Yau [Proc. Symp. Pure Math. None, 619–625 (1991; Zbl 0739.32001)] states that a complete non-compact Kähler manifld with positive bisectional curvature is biholomorphic to \(\mathbb{C}^n\). A special case of this conjecture, namely when the manifold in addition has maximal volume growth, was recently proven by G. Liu [Camb. J. Math. 7, No. 1–2, 33–70 (2019; Zbl 1479.53075)]. The paper under review gives a new proof, using the Kähler-Ricci flow.
The proof uses several ideas and techniques. Most of the paper considers, in fact, the Chern-Ricci flow, which is an analogue of the Kähler-Ricci flow for non-Kähler manifolds. In that setting, the authors prove non-compact analogues of results due to V. Tosatti and B. Weinkove [J. Differ. Geom. 99, No. 1, 125–163 (2015; Zbl 1317.53092)] (characterising the length of time the flow can run) and a few other results, giving a good foundational understanding of the behaviour of the flow.
The results of the paper should be important for the general problem of understanding the geometry of non-compact non-Kähler complex manifolds.

MSC:

32Q15 Kähler manifolds
53C55 Global differential geometry of Hermitian and Kählerian manifolds
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