Collapsing of the ChernRicci flow on elliptic surfaces
Abstract
We investigate the ChernRicci flow, an evolution equation of Hermitian metrics generalizing the KahlerRicci flow, on elliptic bundles over a Riemann surface of genus greater than one. We show that, starting at any Gauduchon metric, the flow collapses the elliptic fibers and the metrics converge to the pullback of a KahlerEinstein metric from the base. Some of our estimates are new even for the KahlerRicci flow. A consequence of our result is that, on every minimal nonKahler surface of Kodaira dimension one, the ChernRicci flow converges in the sense of GromovHausdorff to an orbifold KahlerEinstein metric on a Riemann surface.
 Publication:

arXiv eprints
 Pub Date:
 February 2013
 arXiv:
 arXiv:1302.6545
 Bibcode:
 2013arXiv1302.6545T
 Keywords:

 Mathematics  Differential Geometry;
 53C44;
 53C55;
 32W20
 EPrint:
 46 pages, final version with minor changes to the presentation, to appear in Math. Ann