$SU(2)$ Yang-Mills-Higgs functional with Higgs self-interaction on $3$-manifolds
Da Rong Cheng, Daniel Fadel, Luiz Lara
TL;DR
This work analyzes the SU(2) Yang–Mills–Higgs energy on a 3-manifold with an $ rac{\lambda}{4\varepsilon^2}(1-|\Phi|^2)^2$ Higgs potential, constructing nontrivial critical points via a two-parameter min–max scheme in the small-$\varepsilon$ regime. It establishes a robust variational framework with gauge invariance, proves energy bounds and irreducibility under suitable topology, and develops a detailed a priori regularity theory including coarse and exponential decay estimates, leading to local convergence and bubbling analyses. In the small-$\varepsilon$ limit, the energy measures concentrate as a sum of a harmonic density $|h|^2\,d\mathrm{vol}_g$ and point masses $\Theta(x)\delta_x$ at a finite set $S$, with each $\Theta(x)$ arising from $\mathcal{Y}_1$-critical points on $\mathbb{R}^3$, i.e., finite-energy nonabelian bubbles. The results also yield an energy gap for irreducible solutions on manifolds with bounded geometry and, as a byproduct, guarantee nontrivial $\mathcal{Y}_1$-critical points on $\mathbb{R}^3$ for any $\lambda>0$, enriching the nonabelian monopole theory and suggesting directions toward higher-dimensional gamma-convergence and bubble-tree descriptions.
Abstract
Fixing a constant $λ>0$, for any parameter $\varepsilon>0$ we study critical points of the Yang--Mills--Higgs energy \[ \mathcal{Y}_{\varepsilon}(\nabla,Φ) = \int_M \varepsilon^2|F_{\nabla}|^2 + |\nablaΦ|^2 + \fracλ{4\varepsilon^2}(1-|Φ|^2)^2, \] defined for pairs $(\nabla,Φ)$, where $\nabla$ is a connection on an $SU(2)$-bundle over an oriented Riemannian $3$-manifold $(M^3, g)$, and $Φ$ a section of the associated adjoint bundle. When $M$ is closed, we use a $2$-parameter min-max construction to produce, for $\varepsilon\ll_M 1$, non-trivial critical points in the energy regime \[ 1 \lesssim_λ\varepsilon^{-1}\mathcal{Y}_{\varepsilon}(\nabla_{\varepsilon},Φ_{\varepsilon}) \lesssim_{λ, M} 1. \] When $b_1(M)=0$, these critical points are irreducible: $\nabla_{\varepsilon}Φ_{\varepsilon}\neq 0$. Next, assuming $M$ has bounded geometry (not necessarily compact), and given critical points with $\varepsilon^{-1}\mathcal{Y}_{\varepsilon}(\nabla_{\varepsilon}, Φ_{\varepsilon})$ uniformly bounded, we show that as $\varepsilon\to 0$, the energy measures $\varepsilon^{-1}e_{\varepsilon}(\nabla_{\varepsilon}, Φ_{\varepsilon}) vol_{g}$ converge subsequentially to \[ |h|^2 vol_g + \sum_{x \in S}Θ(x)δ_{x}, \] where $h$ is an $L^2$ harmonic $1$-form, $S$ a finite set and each $Θ(x)$ equals the energy of a finite collection of $\mathcal{Y}_{1}$-critical points on $\mathbb{R}^3$. Finally, the estimates involved also lead to an energy gap for critical points on $3$-manifolds with bounded geometry. As a byproduct of our results, we deduce the existence of non-trivial $\mathcal{Y}_{1}$-critical points over $\mathbb{R}^3$ for any $λ>0$.