A stable manifold theorem for the Yang-Mills gradient flow (Q1813681)

Let \(\nabla_ 0\) be any given Yang-Mills connection over a compact Riemannian manifold \(M\). The authors prove the existence of a (finite- codimensional) stable and a (finite-dimensional) unstable manifold at \(\nabla_ 0\) with respect to the gradient flow for the Yang-Mills functional. An unstable manifold is (by definition) a stable manifold for the backward Yang-Mills gradient flow (YMGF). A stable manifold is a submanifold \(S\) of the space of connections, of a certain Sobolev class, such that if \(\nabla_ 1 \in S\), there exists a unique solution \(\nabla(t)\) to the YMGF equations, starting at \(\nabla_ 1\); moreover \(\nabla(t)\) tends to \(\nabla_ 0\), up to the gauge group action, as \(t\) goes to \(\infty\). The authors prove their results by reducing to an abstract evolution equation, factoring out the gauge group action, and proving the appropriate estimates. Related and partially intersecting results are in another paper by the authors [Differential geometry, Proc. 3rd Int. Symp., Peñiscola/Spain 1988, Lect. Notes Math. 1410, 272-285 (1989; Zbl 0687.58034)].