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

  title={The Yang-Mills flow and the Atiyah-Bott formula on compact K{\"a}hler manifolds},
  author={Adam Jacob},
  journal={American Journal of Mathematics},
  pages={329 - 365}
  • Adam Jacob
  • Published 2011
  • Mathematics
  • American Journal of Mathematics
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… Expand
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