What is known about the Yang-Mills stratification over 3-manifolds?

2012-06-19T18:41:38Z

Rade proved in his thesis (Crelle's Journal, 1992, available here: digizeitschriften.de/dms/toc/?PPN=PPN243919689) that if $E\rightarrow M$ is a $U(n)$ -bundle over a 3-manifold, then the gradient flow of the Yang-Mills functional $L$ on the space $A(E)$ of (Sobolev) connections on $E$ is well-defined and, for any starting connection $A$ , the flow line converges to a critical point (i.e. a Yang-Mills connection) that I'll call $A_\infty$ . In particular, there are no finite time singularities (i.e. no bubbling) as there would be in 4 dimensions. This means that there is a well-defined "stratification" of $A(E)$ given by partitioning $A(E)$ according to the relation $A\sim B$ if $L(A_\infty) = L(B_\infty)$ . If $t\in \mathbb{R}$ is a critical value of $L$ , I'll write $C_t$ for the associated stratum. Rade showed that the gradient flow defines a deformation retraction from $C_t$ to its subset $L^{-1} (t)$ of Yang-Mills connections.

I'm curious if anything at all is known about this stratification. For instance:

Are the subsets $C_t$ actually submanifolds? I believe this would follow from general principals if $L$ satisfied the Palais-Smale Condition C, but it does not. (However, $L$ does satisfy a version of Condition C after modding out gauge transformations; I'm not sure how helpful that is.)
If the $C_t$ are submanifolds, is anything known about their codimensions?

In 2-d Yang-Mills theory, Daskalopoulos provided detailed answers to both questions (building on ideas of Atiyah-Bott). But his arguments make heavy use of complex analytic methods, using the equivalence between Hermitian connections and complex structures in the 2-d case.

Surely one needs to be a little careful about the exact Sobolev regularity of the connections used, but I'm not terribly concerned about that (i.e. I'm happy to assume extra regularity if it helps anything).

What is known about the Yang-Mills stratification over 3-manifolds? - MathOverflow

What is known about the Yang-Mills stratification over 3-manifolds?