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Commentarii Mathematici Helvetici

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Volume 90, Issue 1, 2015, pp. 75–120
DOI: 10.4171/CMH/347

Published online: 2015-02-23

The Yang–Mills $\alpha$-flow in vector bundles over four manifolds and its applications

Min-Chun Hong[1], Gang Tian[2] and Hao Yin[3]

(1) The University of Queensland, Brisbane, Australia
(2) Princeton University, USA
(3) University of Science and Technology of China, Hefei, China

In this paper we introduce an $\alpha$-flow for the Yang-Mills functional in vector bundles over four dimensional Riemannian manifolds, and establish global existence of a unique smooth solution to the $\alpha$-flow with smooth initial value. We prove that the limit of the solutions of the $\alpha$-flow as $\alpha\to 1$ is a weak solution to the Yang-Mills flow. By an application of the $\alpha$-flow, we then follow the idea of Sacks and Uhlenbeck [22] to prove some existence results for Yang-Mills connections and improve the minimizing result of the Yang-Mills functional of Sedlacek [26]

Keywords: Yang–Mills flow, Sacks–Uhlenbeck functional

Hong Min-Chun, Tian Gang, Yin Hao: The Yang–Mills $\alpha$-flow in vector bundles over four manifolds and its applications. Comment. Math. Helv. 90 (2015), 75-120. doi: 10.4171/CMH/347