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arXiv · 2409.18068

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

André Guerra·Xavier Lamy·Konstantinos Zemas
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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.

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André Guerra, Xavier Lamy, Konstantinos Zemas. 2024-09-26. On the existence of degenerate solutions of the two-dimensional $H$-system. https://arxiv.org/abs/2409.18068

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