The Yang-Mills flow and the Atiyah-Bott formula on compact Kähler manifolds

We study the Yang-Mills flow on a holomorphic vector bundle $E$ over a compact K\"ahler manifold $X$. Along a solution of the flow, we show that the curvature endomorphism $i\Lambda F(A_t)$ 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 $E_\infty$ is isomorphic to ${\rm Gr}^{{\rm hns}}(E)^{**}$, verifying a conjecture of Bando and Siu.