全纯平面曲线的复渐屈线
Complex evolutoids of holomorphic plane curves
查看机构详情
- Universidade Federal de Uberlândia（联邦乌贝拉巴大学）
机构由 AI 辅助整理，请以论文原文为准。
浏览论文内容
中文总结 AI 辅助
本文研究 $\mathbb{C}^2$ 中正则全纯曲线的复渐屈线，通过生成族积分导出波前参数化，分类局部奇点，并证明 $A_3$ 燕尾奇点的通用展开条件，揭示各向同性点导致通用性破坏。
中文摘要 AI 辅助
我们研究了在配备标准复双线性度量的 $\mathbb{C}^2$ 中正则全纯曲线的复渐屈线的局部几何、奇点分类和波前动力学。通过积分旋转复仿射线族的生成族，我们导出了 1 参数复波前族 $\Gamma^{\theta, w}$ 的显式参数化，并确立了其尖点奇点轨迹扫出复渐屈线 $E^\theta$。利用无平方根公式，我们对 $E^\theta$ 的局部行为进行分类，证明了渐屈线在各向同性点（$q=0$）处保持正则并与曲线相切，而在复拐点（$\kappa=0$）处逃逸至无穷。最后，我们分析了由演化族 $E^\theta$ 形成的三维复判别曲面 $\mathcal{D}_F \subset \mathbb{C}^3$。我们证明了 $A_2$ 尖点由空间参数通用展开，而 $A_3$ 燕尾奇点由 $(x_1, x_2, \theta)$ 全纯通用展开当且仅当复非退化条件 $\kappa^4 + \kappa_s^2 \neq 0$ 成立。关键的是，我们证明了简单各向同性点导致该通用性条件的破坏，从而诱导复燕尾曲面的退化非横截截面。
英文摘要
We study the local geometry, singularity classification, and wavefront dynamics of complex evolutoids for regular holomorphic curves in $\mathbb{C}^2$ equipped with the standard complex bilinear metric. By integrating the generating family of rotated complex affine lines, we derive an explicit parametrization for the 1-parameter family of complex wavefronts $Γ^{θ, w}$ and establish that their singular cusp locus sweeps out the complex evolutoid $E^θ$. Using a square-root-free formulation, we classify the local behavior of $E^θ$, proving that evolutoids remain regular and tangent to the curve at isotropic points ($q=0$) while escaping to infinity at complex inflections ($κ=0$). Finally, we analyze the three-dimensional complex discriminant surface $\mathcal{D}_F \subset \mathbb{C}^3$ formed by the evolving family $E^θ$. We prove that $A_2$ cusps are versally unfolded by spatial parameters, whereas $A_3$ swallowtail singularities are holomorphically versally unfolded by $(x_1, x_2, θ)$ if and only if the complex non-degeneracy condition $κ^4 + κ_s^2 \neq 0$ holds. Crucially, we demonstrate that simple isotropic points drive the breakdown of this versality condition, inducing degenerate, non-transverse sections of the complex swallowtail surface.
