AI 中文总结
本文提出一种基于表示论的方法，仅利用拓扑与对称群计算嵌入极小曲面的Morse指标与零化度，并应用于自由边界极小曲面及Lawson曲面，结果不依赖具体构造。
AI 中文摘要
我们发展了一种表示论方法，仅基于嵌入极小曲面的拓扑和对称群来确定其Morse指标与零化度。该方法将面积与能量的第二变分之间的等变比较与二面体群的表示论相结合。在欧几里得单位球中，对于每个$k\geq3$，我们计算了具有$4k$阶棱柱对称的任意自由边界极小$k$-noid，以及具有连通边界和八阶反棱柱对称的任意亏格一自由边界极小曲面的Morse指标与零化度。在标准球面中，我们仅利用Lawson曲面第一族对称群的一半，就获得了其Morse指标与零化度的另一种计算方法。这些结果不依赖于曲面的任何特定构造。
英文摘要
We develop a representation-theoretic approach to determine the Morse index and nullity of an embedded minimal surface based only on its topology and symmetry group. The method combines an equivariant comparison between the second variations of area and energy with the representation theory of the dihedral group. In the Euclidean unit ball we compute for every $k\geq3$ the Morse index and nullity of any free boundary minimal $k$-noid with prismatic symmetry of order $4k$, and of any free boundary minimal surface of genus one with connected boundary and antiprismatic symmetry of order eight. In the round sphere we obtain an alternative computation for the Morse index and nullity for the first family of the Lawson surfaces, using only half of their symmetry group. The results are independent of any particular construction of the surfaces.
