Finite time blowing-up for the Yang-Mills gradient flow in higher dimensions
Let \(P\) be a non-trivial principal \(G\)-bundle over \(S^n\) \((n \geq 5)\) with the standard metric \(g\). Yang-Mills connections over \(P\) are critical points of the functional \[ E(D) = {1\over 2} \int_M |F(D)|^2 dV, \] where \(F(D)\) is the curvature for of the connection \(D\). In the paper under review the author considers the Yang-Mills gradient flow \[ {\partial D \over \partial t} = -d^*_D F(D) \quad \text{ on } M \times [0,T),\quad D(0) = D_0,\tag{*} \] where \(d^*_D\) is the formal adjoint operator of the exterior derivative \(d_D\) with respect to \(D\). He proves the following result: There exists a constant \(\varepsilon_1 > 0\), such that if \(D_0\) satisfies \(|F (D_0)|_{L^2 (S^n)} < \varepsilon_1\), then the smooth solution of \((*)\) with initial value \(D_0\) blows up in finite time.
scientific article; zbMATH DE number 720799