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).
