Frobenius conjugation and Jacobi functions for minimal immersions in ${\mathbb{S}}^3$ and CMC $1$ immersions in ${\mathbb{R}}^3$

We give a first-order Frobenius description of conjugate minimal immersions in the round three-sphere and use it to define the conjugate of an arbitrary Jacobi function. The construction is intrinsic at the linearized level and does not require the Jacobi function to be integrable through a normal deformation by minimal surfaces. We then describe how the boundary conditions are exchanged under conjugation. Along a great-circle boundary arc, a Dirichlet Jacobi function has, after a suitable ambient gauge, a conjugate Jacobi function with vanishing Neumann data. Conversely, along a reflection curve, vanishing Neumann data yield vanishing Dirichlet data for the conjugate modulo a Jacobi–Killing function. We also construct conjugate Jacobi functions for conjugate CMC~$1$ immersions in ${\mathbb{R}}^3$ and exhibit cylindrical counterexamples to both pure boundary exchanges, even modulo ambient Killing fields.