THE HERMITIAN CURVATURE FLOW ON MANIFOLDS WITH NON-NEGATIVE GRIFFITHS CURVATURE

In this paper we study a particular version of the Hermitian curvature flow (HCF) over a compact complex Hermitian manifold (M, g, J). We prove that if the initial metric has Griffiths positive (non-negative) Chem curvature Omega, then this property is preserved along the flow. On a manifold with Griffiths non-negative Chern curvature the HCF has nice regularization properties, in particular, for any t > 0 the zero set of Omega(xi, (xi) over bar, eta, (eta) over bar) becomes invariant under certain torsion-twisted parallel transport.