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) Chern curvature $Ω$, 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 $Ω(ξ,\barξ,η,\barη)$ becomes invariant under certain torsion-twisted parallel transport.

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