On the existence of degenerate solutions of the two-dimensional $H$-system - Guerra, André; Lamy, Xavier; Zemas, Konstantinos (Comptes Rendus. Mathématique, 2025)
We consider entire solutions $ω \in \dot{H}^1(\mathbb{R}^2;\mathbb{R}^3)$ of the $H$-system \begin{equation*} Δ ω =2ω _x\wedge ω _y\, \end{equation*} which we refer to as bubbles. Surprisingly, and contrary to conjectures raised in the literature, we find that bubbles with degree at least three can be degenerate: the linearized $H$-system around a bubble can admit solutions that are not tangent to the smooth family of bubbles. We then give a complete algebraic characterization of degenerate bubbles.