The gradient flow of Higgs pairs

Abstract

In this paper, we consider the gradient flow of the Yang-Mills-Higgs functional of Higgs pairs on a Hermitian vector bundle (E , H_{0}) over a Kähler surface (M , \omega ) , and study the asymptotic behavior of the heat flow for Higgs pairs at infinity. The main result is that the gradient flow with initial condition (A_{0} , \phi_{0}) converges, in an appropriate sense which takes into account bubbling phenomena, to a critical points (A_{\infty } , \phi_{\infty}) of this functional. We also prove that the limiting Higgs pair (A_{\infty }, \phi_{\infty}) can be extended smoothly to a vector bundle E_{\infty } over (M , \omega ) , and the isomorphism class of the limiting Higgs bundle (E_{\infty } , A_{\infty } , \phi_{\infty}) is given by the double dual of the graded Higgs sheaves associate to Harder-Narasimhan-Seshadri filtration of the initial Higgs bundle (E , A_{0} , \phi_{0}) .