$SU(2)$ Yang-Mills-Higgs functional with Higgs self-interaction on $3$-manifolds

Abstract

Given a constant $\lambda >0$, for any parameter $\epsilon >0$ we study critical points of the Yang--Mills--Higgs energy \[ \mathcal{Y}_{\varepsilon}(\nabla,Φ) = \int_M \left(\varepsilon^2|F_{\nabla}|^2 + |\nablaΦ|^2 + \fracλ{4\varepsilon^2}(1-|Φ|^2)^2\right)\mathrm{vol}_g, \] defined for pairs $(\nabla ,\Phi )$, where $\nabla$ is a connection on an $SU(2)$-bundle over an oriented Riemannian $3$-manifold $(M^3,g)$, and $\Phi$ a section of the associated adjoint bundle. When $M$ is closed, we use a $3$-parameter min-max construction to produce, for $\epsilon {\ll }_{M}1$, 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: ${\nabla }_{\epsilon }{\Phi }_{\epsilon }\ne 0$. Next, assuming $M$ has bounded geometry (not necessarily compact), and given critical points with ${\epsilon }^{-1}{\mathcal{Y}}_{\epsilon }({\nabla }_{\epsilon },{\Phi }_{\epsilon })$ uniformly bounded, we show that as $\epsilon \to 0$, the energy measures ${\epsilon }^{-1}{e}_{\epsilon }({\nabla }_{\epsilon },{\Phi }_{\epsilon }){\mathrm{vol}}_g$ converge subsequentially to \[ |h|^2 \mathrm{vol}_g + \sum_{x \in S}Θ(x)δ_{x}, \] where $h$ is an $L^2$ harmonic $1$-form, $S$ a finite set and each $\Theta (x)$ 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 $\lambda >0$.

arXiv comment115 pages. v2: Minor corrections and clarifications, including the description of the min-max construction and the presentation of the stress-energy tensor preceding Lemma 5.28. References and acknowledgments updated. Main results unchanged