Communications in Analysis and Geometry

Volume 28 (2020)

Number 6

A new geometric flow over Kähler manifolds

Pages: 1251 – 1288



Yi Li (School of Mathematics and Shing-Tung Yau Center, Southeast University, Nanjing, China)

Yuan Yuan (Department of Mathematics, Syracuse University, Syracuse, New York, U.S.A.)

Yuguang Zhang (Institute of Differential Geometry, Leibniz Universität Hannover, Germany)


In this paper, we introduce a geometric flow for Kähler metrics $\omega_t$ coupled with closed $(1,1)$‑forms $\alpha_t$ on a compact Kähler manifold, whose stationary solution is a constant scalar curvature Kähler (cscK) metric, coupled with a harmonic $(1,1)$‑form. We establish the long-time existence, i.e., assuming the initial $(1,1)$‑form $\alpha$ is nonnegative, then the flow exists as long as the norm of the Riemannian curvature tensors are bounded.

The first author is supported in part by the Fonds National de la Recherche Luxembourg (FNR) under the OPEN scheme (project GEOMREV O14/7628746); the second author is supported in part by NSF grant DMS-1412384; and the third author is supported in part by the Simons Foundation’s program: Simons Collaboration on Special Holonomy in Geometry, Analysis and Physics.

Received 15 October 2015

Accepted 5 February 2018

Published 2 December 2020