通过覆盖空间解决普拉托问题
Plateau's Problem via covering spaces
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中文总结 AI 辅助
研究通过覆盖空间解决普拉托问题，扩展布拉克构造到所有正规子群，证明紧致性结果找到最小面积的\(\Sigma_{N_0}\)，并研究了\(\Sigma_N\)和\(\Sigma_{N_0}\)的相关性质。
中文摘要 AI 辅助
1995年，布拉克提出了一种针对给定边界\(\Gamma\)的普拉托问题的公式，该公式允许三重交点和四面体奇点。设\(\Gamma\)为光滑闭曲线，\(G = \pi_1(\mathbb{R}^3\setminus \Gamma)\)。对于\(G\)的每个真有限指数子群\(N\)，布拉克构造了一个\((\mathrm{\mathbf{M}}, 0, \infty)\)-极小曲面\(\Sigma_N\)，它是与\(N\)相关的覆盖空间中周长最小的基本域边界的投影。我们从两方面推进该理论。一方面，将布拉克的构造扩展到包括所有正规子群\(N\triangleleft G\)，即可能具有无限指数。另一方面，证明了一个紧致性结果，意味着存在一个真正规子群\(N_0\triangleleft G\)，使得\(\mathrm{Area}(\Sigma_{N_0})=\inf_{\{N\triangleleft G, N \neq G\}}\ \mathrm{Area}(\Sigma_N)\)。当\(\Gamma\)有多个连通分量时也有类似结果。此外，我们研究了\(\Sigma_N\)和\(\Sigma_{N_0}\)的跨越和最小化性质。
英文摘要
In 1995, Brakke proposed a formulation of the Plateau problem for a given boundary $Γ$ that allows for triple junctions and tetrahedral singularities. Let $Γ$ be a smooth closed curve and let $G=π_1(\mathbb{R}^3\setminus Γ)$. For each proper finite index subgroup $N$ of $G$, Brakke constructs a $(\mathrm{\mathbf{M}}, 0, \infty)$-minimal surface $Σ_N$, which is obtained as the projection of the boundary of a perimeter-minimising fundamental domain in the covering space associated to $N$. We advance the theory in two ways. Firstly, we extend Brakke's construction to include all normal subgroups $N\triangleleft G$, i.e. possibly with infinite index. Secondly, we prove a compactness result which implies that there exists a proper normal subgroup $N_0\triangleleft G$ such that \[ \mathrm{Area}(Σ_{N_0})=\inf_{\{N\triangleleft G, N \neq G\}}\ \mathrm{Area}(Σ_N). \] A similar result holds when $Γ$ has many connected components. Furthermore, we study the spanning and minimising properties of the $Σ_N$s and $Σ_{N_0}$.
