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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.

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Here is a counterexample.

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 $\mathbb{R} \times Y$.

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    $\begingroup$ Oops, just saw this. I like this argument, but I don't think it's an honest counterexample to the problem in the way that I intended? The boundary condition on the submanifold X' is in this case allowed to depend on the ambient manifold, so this argument does not preclude the case of a boundary condition on B^4 for the psc manifold X' such that M(B^4, P) extends to M(X'). In fact, I could just take the empty boundary condition on B^4 and I believe in that case M(B^4) is also empty? $\endgroup$ Dec 22, 2017 at 17:49
  • $\begingroup$ 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. $\endgroup$
    – Piotr
    Dec 26, 2017 at 15:22
  • $\begingroup$ Secondly, if the boundary condition is allowed to depend on the ambient manifold, one can take the "boundary condition" to be "things that extend to the ambient manifold", right? Also, in such case, replacing the "exterior" would change the boundary conditions, and I don't see how one could expect any gluing formulas... $\endgroup$
    – Piotr
    Dec 26, 2017 at 15:26

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