I will talk about recent joint work with Daniel Fadel (University of São Paulo) and Luiz Lara (Unicamp), where we study the SU(2) Yang-Mills-Higgs functional with positive coupling constant on 3-manifolds. Motivated by the work of Alessandro Pigati and Daniel Stern (2021) on the U(1)-version of the functional, we also include a scaling parameter.

When the 3-manifold is closed and the parameter is small enough, by adapting to our context the min-max method used by Pigati and Stern, we construct non-trivial critical points satisfying energy upper and lower bounds that are natural from the point of view of scaling.

Then, over 3-manifolds with bounded geometry, we show that, in the limit as the parameter tends to zero, and under the above-mentioned energy upper bound, a sequence of critical points exhibits concentration phenomenon at a finite collection of points, while the remaining energy goes into an $L^2$ harmonic 1-form. Moreover, the concentrated energy at each point is accounted for by finitely many "bubbles", that is, non-trivial critical points on $R^3$ with the scaling parameter set to 1.