保持对称性的常中曲率曲面的变形与Hopf微分
Deformations of constant mean curvature surfaces preserving symmetries and the Hopf differential
摘要：
定义了欧几里得空间$E^3$中保持Hopf微分的极小和非极小常中曲率（CMC）曲面之间的变形.我们证明，给定一个CMC $H$表面$f$，无论是最小的或没有，和一个固定的基点$z_0$在这个表面上，有一个自然定义的家庭$f_h$，所有真正的$h$，CMC $h$表面是相切的$f$在$z_0$，并具有相同的霍普夫微分。在给定极小曲面的经典Weierstrass数据的基础上，给出了非极小曲面$f_h$的广义Weierstrass数据的显式表达式，反之亦然.作为一个应用，我们用它来给一个定义良好的修饰行动类的极小曲面。此外，我们表明，与基点相关的某些类型的对称性被保存下的变形，这给出了一个典型的选择与对称性的表面的基点。我们用它来定义新的非极小CMC表面的例子自然与已知的极小曲面的对称性。
英文摘要：
We define certain deformations between minimal and non-minimal constant mean curvature (CMC) surfaces in Euclidean space $E^3$ which preserve the Hopf differential. We prove that, given a CMC $H$ surface $f$, either minimal or not, and a fixed basepoint $z_0$ on this surface, there is a naturally defined family $f_h$, for all real $h$, of CMC $h$ surfaces that are tangent to $f$ at $z_0$, and which have the same Hopf differential. Given the classical Weierstrass data for a minimal surface, we give an explicit formula for the generalized Weierstrass data for the non-minimal surfaces $f_h$, and vice versa. As an application, we use this to give a well-defined dressing action on the class of minimal surfaces. In addition, we show that symmetries of certain types associated with the basepoint are preserved under the deformation, and this gives a canonical choice of basepoint for surfaces with symmetries. We use this to define new examples of non-minimal CMC surfaces naturally associated to known minimal surfaces with symmetries.
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