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THE ALMOST HERMITIAN-EINSTEIN FLOWS

Let X be an n-dimensional compact Kähler manifold, E → X a Hermitian vector bundle and L → X an ample line bundle. We construct a family of evolution equations corresponding to the almost Hermitian-Einstein equations introduced by Leung in [2] and [3] and prove that the solution exists for a short time. We consider a family of funtionals Dη corresponding to the family of evolution equations and show that each one of these functional is decreasing along the corresponding evolution equation. We also show that each one of these functionals is a potential funtion on the space of Hermitian metrics on E.