THE YANG-MILLS FLOW AND THE ATIYAH-BOTT FORMULA ON COMPACT KAHLER MANIFOLDS

We study the Yang-Mills flow on a holomorphic vector bundle E over a compact Kahler manifold X. Along a solution of the flow, we show that the curvature endomorphism iAF(A(t)) approaches in L-2 an endomorphism with constant eigenvalues given by the slopes of the quotients from the Harder-Narasimhan filtration of E. This proves a sharp lower bound for the Hermitian-Yang-Mills functional and thus the Yang-Mills functional, generalizing to arbitrary dimension a formula of Atiyah and Bott first proven on Riemann surfaces. Furthermore, we show any reflexive extension to all of X of the limiting bundle E is isomorphic to Gr(hns)(E)**, verifying a conjecture of Bando and Siu.