Solutions of the Bogomolny Equation on R^3 with Certain Type of Knot Singularity I
Weifeng Sun
TL;DR
This work analyzes the moduli space of solutions to the Bogomolny equations on $\mathbb{R}^3 \setminus K$ with a knot singularity, introducing a knot-monodromy parameter $\gamma$, a mass $M$ and a charge $k$ at infinity. The author develops a Fredholm theory for the linearization via carefully chosen weighted Sobolev spaces $\mathbb{H}$ and $\mathbb{L}$, constructs a global model by gluing a knot-model to infinity-model configurations, and proves existence of actual solutions through a gluing/implicit-function framework for all $\gamma \neq \tfrac{1}{4}$. Regularity results near the knot and asymptotic analysis at infinity yield a detailed description of solution behavior, including a Uhlenbeck-type regularity theory and a precise mass/charge asymptotics, with the moduli space formed as an analytic set modulo a 1-dimensional gauge ambiguity. The construction suggests potential applications in low-dimensional topology and knot theory via moduli spaces of monopoles with knot singularities. Mathematical structures like edge calculus and Taubes-style gluing underpin the approach, linking gauge theory with knot singularities in $\mathbb{R}^3$.
Abstract
Moduli space of the Bogomolny equation on R^3 with certain asymptotic conditions at infinity has been well studied for a long time. This paper studies the moduli space of solutions to the Bogomolny equation on R^3 with a knot singularity. The author hopes such kind of moduli spaces have potential applications in low-dimensional topology and knot theory in the future.