On the existence of degenerate solutions of the two-dimensional $H$-system

TL;DR

This paper investigates entire solutions of the two-dimensional $H$-system, revealing that bubbles with degree three or higher can be degenerate, contrary to previous beliefs, and provides a full algebraic classification of such degenerate solutions.

Contribution

It demonstrates the existence of degenerate bubbles with degree at least three and offers a comprehensive algebraic characterization of these degenerate solutions.

Findings

Bubbles with degree ≥ 3 can be degenerate.

Linearized $H$-system admits non-tangent solutions around certain bubbles.

Complete algebraic classification of degenerate bubbles provided.

Abstract

We consider entire solutions $\omega\in\dot H^1(\mathbb R^2;\mathbb R^3)$ of the $H$-system $\Delta\omega=2\omega_x\wedge\omega_y,$ 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.