In this paper, we consider the gradient flow of the Yang-Mills-Higgs functional of Higgs pairs on a Hermitian vector bundle over a Kähler surface , 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 converges, in an appropriate sense which takes into account bubbling phenomena, to a critical points of this functional. We also prove that the limiting Higgs pair can be extended smoothly to a vector bundle over , and the isomorphism class of the limiting Higgs bundle is given by the double dual of the graded Higgs sheaves associate to Harder-Narasimhan-Seshadri filtration of the initial Higgs bundle .
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Jiayu Li, Xi Zhang, The gradient flow of Higgs pairs. J. Eur. Math. Soc. 13 (2011), no. 5, pp. 1373–1422DOI 10.4171/JEMS/284