Limiting measures and energy growth for sequences of solutions to Taubes's Seiberg-Witten equations

TL;DR

This paper investigates the asymptotic behavior of solutions to Taubes's modified Seiberg-Witten equations on 3-manifolds, especially when the energy is unbounded, revealing new convergence properties and measure restrictions.

Contribution

It introduces a maximum principle applicable to unbounded energy solutions, providing new insights into the limiting behavior and measures of solutions with unbounded energy growth.

Findings

Limiting nodal sets converge to invariant sets of the vector field when energy grows slower than r_n^{1/2}

New maximum principle valid for solutions with unbounded energy

Unbounded energy solutions have no local restrictions on limiting measures

Abstract

We consider sequences of solutions $(\psi_n,A_n)_{n=1}^\infty$ to Taubes's modified Seiberg-Witten equations, associated with a fixed volume-preserving vector field $X$ on a 3-manifold and corresponding to arbitrarily large values of the strength parameter $r_n \to \infty$. In Taubes's work, the asymptotic behavior of these solutions is related to the dynamics of $X$. We consider the rather unexplored case of sequences of solutions whose energy is not uniformly bounded as $n\to\infty$. Our first main result shows that when the energy grows more slowly than $r_n^{1/2}$, the limiting nodal set of the solutions converges to an invariant set of the vector field $X$. The main tool we use is a novel maximum principle for the solutions with the key property that it remains valid in the unbounded energy case. As a byproduct, in the usual case of sequences of solutions with bounded energy, we obtain a new, more straightforward proof of Taubes's result on the existence of periodic orbits that does not involve a local analysis or the vortex equations. Our second main result proves that, contrary to what happens in the bounded energy case, when the energy is unbounded there are no local restrictions to the limiting measures that may arise in the modified Seiberg-Witten equations. Furthermore, we obtain a connection about the dimension of the support of the limiting measure (as expressed through a $d$-Frostman property) and the energy growth of the sequence of local solutions we construct.