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

2024/09/26 by Guerra, André, Lamy, Xavier, Zemas, Konstantinos
#Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2409.18068

Abstract

We consider entire solutions ω∈ H¹(\mathbb R²;\mathbb R³) of the H-system Δω=2ω_x\wedgeω_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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