发表机构
Texas Tech University（德克萨斯理工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明单位球中具有棱柱对称性的亏格零自由边界极小曲面的Morse指标为2b，零化度为三，并给出首批精确指标的例子及对称性指标下界。
AI 中文摘要
对于整数$b\ge3$，设$\Sig\subset\Ball$为具有$b$个边界分量、亏格为零的嵌入自由边界极小曲面（FBMS）。假设$\Sig$在阶为$4b$的棱柱群下不变，其中阶为$b$的旋转循环地置换边界分量，而赤道平面中的反射保持每个边界分量不变。我们证明$\Sig$的Morse指标为$2b$，其零化度为三；此外，我们将指标形式的负空间确定为对称群的表示。例子包括Karpukhin、Kusner、McGrath和Stern的$b$-noids，以及对于大的$b$，Folha、Pacard和Zolotareva的亏格为零曲面。据我们所知，除了赤道圆盘和临界悬链面之外，这些是球中首批Morse指标被精确知道的FBMS。对于具有旋转或多面体对称性的曲面，也有指标的下界。主要的新工具是使用有限群表示论的等变比较，在面积与能量之间进行，其中Teichmüller方向通过圆域按对称类型计数。
英文摘要
For integer $b\ge3$, let $\Sig\subset\Ball$ be an embedded free boundary minimal surface (FBMS) of genus zero with $b$ boundary components. Suppose that $\Sig$ is invariant under the prismatic group of order $4b$, the rotation of order $b$ permutes the boundary components cyclically, and the reflection in the equatorial plane preserves each of them. We show that the Morse index of $\Sig$ is $2b$ and its nullity is three; moreover, we determine the negative space of the index form as a representation of the symmetry group. Examples include the $b$-noids of Karpukhin, Kusner, McGrath, and Stern and, for large $b$, the genus-zero surfaces of Folha, Pacard, and Zolotareva. To our knowledge, apart from the equatorial disc and the critical catenoid, these are the first FBMS in the ball whose Morse indices are known precisely. There are also lower bounds for the index of surfaces with rotational or polyhedral symmetry. The main new ingredient is an equivariant comparison, using the representation theory of finite groups, between area and energy, in which the Teichmüller directions are counted by symmetry type using circle domains.
Comments31 pages, 2 figures
