AI 中文总结
本文用一阶 Frobenius 描述定义极小浸入的共轭 Jacobi 函数，研究边界条件交换，并给出 CMC 1 浸入的构造及反例。
AI 中文摘要
我们给出了圆三球面中共轭极小浸入的一阶 Frobenius 描述，并利用它定义了任意 Jacobi 函数的共轭。该构造在线性化层面是内蕴的，不要求 Jacobi 函数可通过极小曲面的法向形变积分。随后我们描述了边界条件在共轭下的交换。沿大圆边界弧，Dirichlet 型 Jacobi 函数在适当的环境规范下，其共轭 Jacobi 函数具有消失的 Neumann 数据。反之，沿反射曲线，消失的 Neumann 数据导致共轭的 Dirichlet 数据模一个 Jacobi--Killing 函数为零。我们还为 $\mathbb{R}^3$ 中共轭 CMC $1$ 浸入构造了共轭 Jacobi 函数，并给出了圆柱反例，表明即使模环境 Killing 场，两种纯边界交换也不成立。
英文摘要
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.
Comments67 pages, no figures. Invited contribution to a special issue in Pure and Applied Mathematics Quarterly, in memory of William H. Meeks III
