平移曲面的短同调基
Short homology bases for translation surfaces
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中文总结 AI 辅助
本文证明闭平移曲面存在长度受对数界控制的短同调基，利用 Voronoi 图及其对偶图构造，并给出依赖最短鞍连接长度的补充结果。
中文摘要 AI 辅助
设 $S$ 为亏格 $g\ge 2$ 的闭平移曲面，其面积 $\mathop{area}(S)= 4\pi g$。我们证明：对任意 $\lambda \in (0, 1)$，存在 $\lfloor \lambda \cdot g\rfloor$ 条同调无关的简单闭曲线，其长度至多为 $ C(\lambda) \cdot \log(g)$，其中 $C(\lambda)$ 是仅依赖于 $\lambda$ 的常数。该结果通过基于以锥点为种子的曲面 Voronoi 图及其对偶图的混合图构造获得。我们还给出一个仅使用 Voronoi 图的补充结果，可产生构成同调基的 $2g$ 条短环。然而，该构造依赖于曲面 $S$ 的最短鞍连接的长度。
英文摘要
Let $S$ be a closed translation surface of genus $g\ge 2$ with $\mathop{area}(S)= 4πg$. We show that for any $λ\in (0, 1)$ there exist $\lfloor λ\cdot g\rfloor$ homologically independent simple closed curves of length at most $ C(λ) \cdot \log(g)$, where $C(λ)$ is a constant that depends only on $λ$. This result is obtained from a mixed graph construction based on a Voronoi graph of the surface with the cone points as seeds and its dual graph. We also give a complementary result using only the Voronoi graph that produces $2g$ short loops that form a homology basis. This construction, however, depends on the length of a shortest saddle connection of the surface $S$.
