论文标题
广义亨内伯格稳定的最小表面
Generalized Henneberg stable minimal surfaces
论文作者
论文摘要
我们通过在$ \ mathbb {r}^3 $中提供一个无限的，有限的，不可定向，稳定的最小表面的无限家族，从而概括了经典的Henneberg最小表面。这些表面可以根据正整数（称为复杂性）的不同，可以将其分为亚家族，该整数基本上测量了分支点的数量。经典的Henneberg Surface $ H_1 $被描述为最简单复杂性$ M = 1 $的子家族中的独特示例，而对于$ M \ geq 2 $ geq 2 $多参数系列。 \ mathbb {n} $的最对称示例$ h_m $的等轴测组对二脑等轴测组$ d_ {2m+2} $（如果$ m $是奇数）或$ d_ {m+1}} {m+1} \ times \ mathbb = $ is if $ is if $ d_ if $ d_（如果$ m $ is osg）是同态的。 Furthermore, for $m$ even $H_m$ is the unique solution to the Björling problem for a hypocycloid of $m+1$ cusps (if $m$ is even), while for $m$ odd the conjugate minimal surface $H_m^*$ to $H_m$ is the unique solution to the Björling problem for a hypocycloid of $2m+2$ cusps.
We generalize the classical Henneberg minimal surface by giving an infinite family of complete, finitely branched, non-orientable, stable minimal surfaces in $\mathbb{R}^3$. These surfaces can be grouped into subfamilies depending on a positive integer (called the complexity), which essentially measures the number of branch points. The classical Henneberg surface $H_1$ is characterized as the unique example in the subfamily of the simplest complexity $m=1$, while for $m\geq 2$ multiparameter families are given. The isometry group of the most symmetric example $H_m$ with a given complexity $m\in \mathbb{N}$ is either isomorphic to the dihedral isometry group $D_{2m+2}$ (if $m$ is odd) or to $D_{m+1}\times \mathbb{Z}_2$ (if $m$ is even). Furthermore, for $m$ even $H_m$ is the unique solution to the Björling problem for a hypocycloid of $m+1$ cusps (if $m$ is even), while for $m$ odd the conjugate minimal surface $H_m^*$ to $H_m$ is the unique solution to the Björling problem for a hypocycloid of $2m+2$ cusps.
