Hermitian Curvature Flow

We define a functional for Hermitian metrics using the curvature of the Chern connection. The Euler-Lagrange equation for this functional is an elliptic equation for Hermitian metrics. Solutions to this equation are related to K"ahler-Einstein metrics, and are automatically K"ahler-Einstein under certain conditions. Given this, a natural parabolic flow equation arises. We prove short time existence and regularity results for this flow, as well as stability for the flow near K"ahler-Einstein metrics with negative or zero first Chern class.