关于 $\mathbb{R}^3$ 中不可定向极小曲面的莫尔斯指标
On the Morse index of nonorientable minimal surfaces in $\mathbb{R}^3$
- Indiana University Bloomington（印第安纳大学布鲁明顿分校）
- Instituto de Matemática Pura e Aplicada（纯粹与应用数学研究所）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文计算了三维欧氏空间中若干不可定向极小曲面的莫尔斯指标，证明Meeks莫比乌斯带指标为2、López克莱因瓶为3，并给出指标分类与下界结果。
AI中文摘要：
我们计算了浸入三维欧氏空间中的若干完备不可定向极小曲面的莫尔斯指标。特别地，我们证明了Meeks极小莫比乌斯带的莫尔斯指标为2，根据Ros和Chodosh-Maximo的先前工作，这是此类不可定向极小曲面可能的最小莫尔斯指标。这是已知的第一个指标为2的极小曲面。我们证明López极小克莱因瓶的莫尔斯指标为3。作为该计算的一个推论，我们扩展了文献中的结果，并观察到浸入三维欧氏空间中总曲率为$8 \pi$的任何完备极小曲面的莫尔斯指标为3。我们还计算了Oliveira族极小莫比乌斯带的莫尔斯指标，以其总曲率表示，并证明每个非负整数都是浸入三维欧氏空间中的某个完备极小曲面的莫尔斯指标。我们证明5是浸入三维欧氏空间中的任何极小定向双覆盖的莫尔斯指标的尖锐下界，且等号由Meeks极小莫比乌斯带的双覆盖取得。最后，我们证明了关于浸入三维欧氏空间中莫尔斯指标为2的完备极小曲面分类的结果，获得了此类曲面的拓扑和几何限制。
英文摘要：
We compute the Morse index of several complete nonorientable minimal surfaces immersed in Euclidean three-space. In particular, we prove that the Meeks minimal Möbius band has Morse index two, which is the least possible Morse index for such nonorientable minimal surfaces, by previous works of Ros and Chodosh-Maximo. This is the first known minimal surface with index two. We show that the López minimal Klein bottle has Morse index three. As a consequence of this computation, we extend results in the literature and observe that any complete minimal surface immersed in Euclidean three-space with total curvature $8 π$ has Morse index three. We also compute the Morse index of the Oliveira family of minimal Möbius bands in terms of their total curvature and show that every non-negative integer is the Morse index of a complete minimal surface immersed in Euclidean three-space. We show that five is a sharp lower bound for the Morse index of any minimal oriented double cover in Euclidean three-space, and equality is attained by the double cover of the Meeks minimal Möbius band. Finally, we prove results towards the classification of complete minimal surfaces with Morse index two immersed in Euclidean three-space, obtaining topological and geometric restrictions on such surfaces.
