The Limit of Large Mass Monopoles

TL;DR

This paper investigates the limiting behavior of large mass monopoles on asymptotically conical 3-manifolds, showing energy concentration along finite sets and establishing the coincidence of zero and blow-up sets.

Contribution

It characterizes the blow-up set for monopoles with unbounded energy, providing bounds on its size and linking it to the zero set of Higgs fields in the large mass limit.

Findings

The blow-up set is finite and its size depends only on the monopole charge.

The zero set coincides with the blow-up set in the large mass limit.

Zero and blow-up sets are finite point sets for sequences with unbounded energy.

Abstract

In this paper we consider $\rm SU(2)$ monopoles on an asymptotically conical, oriented, Riemannian $3$-manifold with one end. The connected components of the moduli space of monopoles in this setting are labeled by an integer called the charge. We analyse the limiting behavior of sequences of monopoles with fixed charge, and whose sequence of Yang--Mills--Higgs energies is unbounded. We prove that the limiting behavior of such monopoles is characterized by energy concentration along a certain set, which we call the blow-up set. Our work shows that this set is finite, and using a bubbling analysis obtain effective bounds on its cardinality, with such bounds depending solely on the charge of the monopole. Moreover, for such sequences of monopoles there is another naturally associated set, the zero set, which consists on the set at which the zeros of the Higgs fields accumulate. Regarding this, our results show that for such sequences of monopoles, the zero set and the blow-up set coincide. In particular, proving that in this "large mass" limit, the zero set is a finite set of points. Some of our work extends for sequences of finite mass critical points of the Yang--Mills--Higgs functional for which the Yang--Mills--Higgs energies are $O(m_i)$ as $i\to\infty$, where $m_i$ are the masses of the configurations.