On Kähler-like almost Hermitian metrics and the almost Hermitian curvature flow

Abstract

We introduce a Kähler-like almost Hermitian metric and an almost balanced metric. We prove that on a Kähler-like almost Hermitian manifold, we have an identity between the first derivative of the torsion $(1,0)$-tensor and the Nijenhuis tensor. By applying the identity, then we figure out what the equivalent condition of being almost balanced on a compact Kähler-like almost Hermitian manifold is. We apply the result to a 2-step nilpotent Lie algebra, and also to the almost Hermitian curvature flow (AHCF). We obtain a lower bound for the scalar curvature along (AHCF). Also we have some results on the monotonicity of the volume along (AHCF) by studying the relation between the volume and the scalar curvature.