Asymptotical behaviour of the Yang-Mills flow and singular Yang-Mills connections (Q707412)

The authors analyze the asymptotic behaviour of the Yang-Mills heat flow. Their main result is as follows: Let \(E\) be a vector bundle over a compact Riemannian manifold \(M\). Let \(A\) be a global smooth solution of the Yang-Mills flow in \(M\times [0, \infty)\) with smooth initial value \(A_ 0\). Then there exists a sequence \(\{t_ i\}\) such that, as \(t_ i \to \infty\), \(A(x, t_ i)\) converges, modulo gauge transformations, to a Yang-Mills connection \(A\) in smooth topology outside a closed set \(\Sigma\). Moreover, the set \(\Sigma\) is \((m-4)\)-rectifiable. Next, they generalize this result to the Yang-Mills-Higgs flow and combining the result with the results of Donaldson they obtain a counterpart of their theorem for a holomorpic bundle over a Kähler manifold.