全纯圆盘的Bishop族：正则性与高阶指标
The Bishop family of holomorphic discs: regularity and higher index
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中文总结 AI 辅助
本文研究Bishop全纯圆盘族的正则性，通过实曲面爆破方法解决复点问题，证明非脐椭圆复点处的正则性及高指标复点附近全纯圆盘的存在正则性，相关问题为弗雷德霍姆正则。
中文摘要 AI 辅助
我们证明，对于边界位于$C^{k,\alpha}$正则实曲面上的全纯圆盘的Bishop族，在非脐椭圆复点处具有$C^{k/2,\alpha/2}$正则性。此外，我们证明在某些指标$\ge 2$的复点附近全纯圆盘的存在性与正则性。该证明采用实曲面的新型爆破方法，将复点分解为一对全实曲面，进而导出全纯环带的$\mathbb{Z}_2$等变黎曼-希尔伯特问题。计算得指标为1，且该问题被证明是弗雷德霍姆正则的。
英文摘要
We prove $C^{k/2,α/2}$ -regularity up to a non-umbilic elliptic complex point for the Bishop family of holomorphic discs with boundary in a $C^{k,α}$ regular real surface. Furthermore, we prove existence and regularity of holomorphic discs near certain complex points of index $\ge 2$. The proof employs a novel blow-up of the real surface which resolves the complex point to a pair of totally real surfaces and leads to a $\mathbb{Z}_2$ -equivariant Riemann-Hilbert problem for holomorphic annuli. The index is computed to be 1 and the problem is shown to be Fredholm-regular.
