| The J-flow is a parabolic flow on compact Kahler manifolds with two given Kahler metrics. It is shown that the flow converges to a critical metric if their classes satisfy a certain inequality. A corollary of this is a lower bound for the Mabuchi energy on a specific open neighbourhood of the Kahler cone for manifolds with negative first Chern class.; A complex Frobenius theorem is proved for subsheaves of holomorphic vector bundles. A notion of multiplier ideal sheaf is defined for a sequence of Hermitian metrics. It is shown that the destabilizing subsheaf of Uhlenbeck and Yau can be constructed as such a multiplier ideal sheaf, making use of the complex Frobenius theorem and the Yang-Mills flow.; Also, the formation of singularities for the Yang-Mills flow over a Riemannian manifold is discussed. It is shown that a sequence of blow ups of a Type I singularity converges, modulo the gauge group, to a homothetically shrinking soltion. Examples of such solitons are given for trivial bundles over Rn for 5 ≤ n ≤ 9. |
