Morse 指标、拓扑及具有非紧自由边界的极小曲面的端部
Morse index, topology, and ends of minimal surfaces with noncompact free boundary
- Universidade Federal de Alagoas（阿拉戈阿斯联邦大学）
- Universidade Estadual de Alagoas（阿拉戈阿斯州立大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文为具有非紧自由边界的完备双侧极小曲面建立 Morse 指标下界，结合局部能量恒等式与 Riemann--Roch 计数，并处理内部端与边界端的截断问题，推广了紧边界情形的结果。
AI中文摘要：
我们为光滑平均凸域 $\mathbb{R}^3$ 中具有非紧边界的完备双侧自由边界极小曲面建立了指标估计。首先证明 $3{\rm Ind}_s(\Sigma)\geq 2g+b-1$，其中 $g$ 和 $b$ 描述共形紧化。随后，我们在不附加进一步渐近假设的情况下，计入所有内部端及其重数，并在确保截断误差消失的显式条件下选取边界端。证明结合了局部能量恒等式与共形双上的 Riemann--Roch 计数。若所选形式在边界奇点处正则，则该奇点消除例外空间；若允许在每个边界奇点处有极点，则该空间维数至多为一。我们给出了混合截断构造、区分嵌入边界端与反射平面端的示例，以及对共形 Jacobi 度量的独立分析。后者给出对数共形因子的有限 Dirichlet 能量，但无限制的边界端估计仍是一个开放步骤。紧边界情形已由 Cavalcante、Mendes 和 dos Santos 处理。
英文摘要：
We establish index estimates for complete two-sided free boundary minimal surfaces in smooth mean-convex domains of $\mathbb{R}^3$ with noncompact boundary. We first prove $3{\rm Ind}_s(Σ)\geq 2g+b-1$, where $g$ and $b$ describe the conformal compactification. We then include all interior ends with their multiplicities, without further asymptotic assumptions, and selected boundary ends under an explicit condition ensuring vanishing cutoff errors. The proof combines a localized energy identity with a Riemann--Roch count on the conformal double. A boundary puncture at which the chosen forms are regular eliminates the exceptional space; if poles are allowed at every boundary puncture, this space has dimension at most one. We provide the mixed cutoff construction, examples distinguishing embedded boundary ends from reflected planar ends, and a separate analysis of the conformal Jacobi metric. The latter yields finite Dirichlet energy of the logarithmic conformal factor, but the unrestricted boundary-end estimate remains an open step. The compact-boundary case was treated by Cavalcante, Mendes, and dos Santos.
