# Extension problem for Seiberg-Witten solutions

Let $X$ be a compact $4$-manifold, possibly with boundary.

Theorem 17.1.2 of Kronheimer-Mrowka's book "Monopoles and Three-Manifolds" states

Let $X' \subset X$ be a codimension-zero submanifold with boundary, contained in the interior of the manifold $X$. The restriction map $L^2_k(X) \to L^2_k(X')$ is a surjective map with continuous left inverse.

I was wondering if such an extension statement for solutions to the Seiberg-Witten equations could be found anywhere in the literature.

Specifically, can we fix boundary conditions $P$, $P'$ on $X$, $X'$ such that:

1. There is a well-defined restriction map from $\mathcal{M}(X, P)$ to $\mathcal{M}(X', P')$, the spaces in question being the gauge-equivalence classes of solutions to the Seiberg-Witten equations satisfying the boundary conditions.

2. This restriction map is surjective and admits a continuous left inverse.

Consider a flat ball $B^4$ embedded in a closed manifold $X$ on which there are irreducible solutions to the Seiberg-Witten equations (e.g. a symplectic manifold). Suppose your conjecture is true for some boundary conditions $P_B$ on $\partial B^4$ (there are no boundary conditions on $X$, obviously). Thus $\mathcal{M}(B^4,P_B)$ is nonempty and, specifically, contains an irreducible solution.
Now embed $B^4$ in any closed connected $4$-manifold $X'$ which has nonnegative, somewhere positive scalar curvature. By the Weitzenbock formula and a unique continuation theorem for the Dirac equation, there are no irreducible solutions on $X'$, so we cannot extend the irreducibles from $\mathcal{M}(B^4,P_B)$ to $\mathcal{M}(X',\varnothing)$.
Another (less rigorous) argument: the Seiberg-Witten equation may blow up in finite time on a cylinder $[a,b] \times Y$. Your conjecture would imply existence of some boundary conditions on the cylinder which would guarantee that the solutions do not blow up on $[a-t,b+t] \times Y$ for any $t$. I suppose there should be embeddings of $[a,b] \times Y$ into closed $4$-manifolds $X$ such that restricting the solutions on $X$ to the cylinder yields solutions which blow up in finite time on $\mathbb{R} \times Y$.
• First of all, with an empty boundary condition the set of solutions is nonempty, as I have proved above: one can embed (flat) $B^4$ in a manifold with a nonzero Seiberg-Witten invariant (and $b^+>1$). I bet one can as well explicitly write down some (many) irreducile solutions. Dec 26, 2017 at 15:22