Existence of nonconcentric solutions in the constant-curvature degenerate case

Determine whether the two-dimensional Lyapunov–Schmidt reduced equation associated with the first Fourier-mode resonance for a concentric circular steady vortex sheet in the constant-curvature disk family has nonconcentric zeros, and hence whether nonconcentric normalized steady vortex-sheet solutions exist locally.

Background

For rotationally symmetric reference metrics and concentric circular sheets, the normalized linearized problem decomposes into independent Fourier modes. In the constant-curvature disk family, the determinant is zero for the first mode and strictly negative for every mode k≥2, so the normalized problem has a two-dimensional kernel and cokernel.

The two first-mode kernel profiles coincide with normal traces of ambient space-form Killing fields. However, those Killing-field flows do not preserve the fixed outer boundary of the unit disk, so the kernel cannot be interpreted as producing a symmetry-generated nonconcentric branch. The paper reduces the unresolved issue to determining whether the resulting two-dimensional finite-dimensional reduced equation has zeros away from the concentric configuration.