Hermitian curvature flow on complex homogeneous manifolds

TL;DR

This paper investigates the Hermitian curvature flow on complex homogeneous manifolds, demonstrating invariance properties, constructing special Einstein metrics, and analyzing the flow's behavior on nilpotent and solvable Lie groups.

Contribution

It introduces a finite-dimensional invariant space of metrics under HCF, constructs HCF-Einstein metrics on complex homogeneous manifolds, and studies flow behavior on nilpotent and solvable groups.

Findings

Invariant metric space under HCF on complex homogeneous manifolds

Construction of HCF-Einstein metrics on G-homogeneous manifolds

Analysis of flow blow-up behavior on nilpotent and solvable Lie groups

Abstract

In this paper we study a version of the Hermitian curvature flow (HCF). We focus on complex homogeneous manifolds equipped with induced metrics. We prove that this finite-dimensional space of metrics is invariant under the HCF and write down the corresponding ODE on the space of Hermitian forms on the underlying Lie algebra. Using these computations we construct HCF-Einstein metrics on $G$-homogeneous manifolds, where $G$ is a complexification of a compact simple Lie group. We conjecture that under the HCF any induced metric on such a manifold pinches towards the HCF-Einstein metric. For a nilpotent or solvable complex Lie group equipped with an induced metric we investigate the blow-up behavior of the HCF.