The Li-Yau-Hamilton Estimate and the Yang-Mills Heat Equation on Manifolds with Boundary
The paper pursues two connected goals. Firstly, we establish the Li-Yau-Hamilton estimate for the heat equation on a manifold $M$ with nonempty boundary. Results of this kind are typically used to prove monotonicity formulas related to geometric flows. Secondly, we establish bounds for a solution $\nabla(t)$ of the Yang-Mills heat equation in a vector bundle over $M$. The Li-Yau-Hamilton estimate is utilized in the proofs. Our results imply that the curvature of $\nabla(t)$ does not blow up if the dimension of $M$ is less than 4 or if the initial energy of $\nabla(t)$ is sufficiently small.
Global existence and convergence of smooth solutions to Yang-Mills gradient flow over compact four-manifolds
In this monograph we develop results on global existence and convergence of solutions to the gradient flow equation for the Yang-Mills energy functional on a principal bundle, with compact Lie structure group, over a closed, four-dimensional, Riemannian, smooth manifold, including the following. If the initial connection is close enough to a minimum of the Yang-Mills energy functional, in a norm or energy sense, then the Yang-Mills gradient flow exists for all time and converges to a Yang-Mills connection. If the initial connection is allowed to have arbitrary energy but we restrict to the setting of a Hermitian vector bundle over a compact, complex, Hermitian (but not necessarily Kaehler) surface and the initial connection has curvature of type $(1,1)$, then the Yang-Mills gradient flow exists for all time, though bubble singularities may (and in certain cases must) occur in the limit as time tends to infinity. The Lojasiewicz-Simon gradient inequality plays a crucial role in our approach and we develop two versions of that inequality for the Yang-Mills energy functional.
Completeness of compact Lorentz manifolds admitting a timelike conformal Killing vector field
It is proved that every compact Lorentz manifold admitting a timelike conformai Killing vector field is geodesically complete. So, a recent result by Kamishima in J. Differential Geometry [37 (1993), 569-601] is widely extended. Recently, it has been proved in [2] that a compact Lorentz manifold of constant curvature admitting a timelike Killing vector field is (geodesically) complete. It is natural to think about the importance of the assumption on the curvature in this result. Moreover, we could ask ourselves if there is a more general condition than the existence of a timelike Killing vector field. The answers to these questions are given in the following Theorem. Let (M, g) be a compact Lorentz manifold which admits a timelike conformai Killing vector field K. Then (M, g) is geodesically complete. Proof. We are going to see that any geodesic y : [0, e[-> M, 0 0. Thus, we have only to check that g(K, y') is bounded. Taking into account the fact that Lng = o • g for some function o , we obtain that j-(g(K, y') = ^C-ooy, so (d/dt)g(K, y') and, as a consequence g(K, y') is bounded on [0, b[. Received by the editors October 20, 1993 and, in revised form, January 25, 1994. 1991 Mathematics Subject Classification. Primary 53C50, 53C22.