哪些完整环绕特征是可实现的？闭曲面的完整答案
Which Holonomy Signatures Are Realizable? A Complete Answer for Closed Surfaces
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中文总结 AI 辅助
该研究解决闭曲面无缝参数化的完整环绕特征可实现性问题，通过归约引理、对应关系结合微分层分类，确定5个例外族外所有特征可实现，给出构造性方法并扩展至多类曲面。
中文摘要 AI 辅助
闭定向曲面的无缝参数化带有一个离散不变量，即其完整环绕特征：所有均为π/2整数倍的锥角，以及诱导交叉场的旋转完整环绕ρ∶H₁(M∖C)→ℤ₄。这是四边形网格所规定的数据，它决定了是否存在任何参数化。Shen、Zhu、Capouellez、Panozzo、Campen和Zorin曾询问哪些特征会出现，并给出了一种gcd型充分条件；哪些特征是可实现的这一问题仍未解决。我们回答了该问题。一个归约定理表明，映射类群在具有固定锥角的特征上的作用，其轨道仅由子群im ρ≤ℤ₄分类，因此每组角多重集最多有3种情况，而非4²ᵍ种。随后的对应关系将无缝参数化与亚纯4-微分对应起来，其中im ρ衡量本原性，可实现性则转化为本原k-微分层的非空性，k=4/d且im ρ=⟨d⟩。结合这类层的已知分类，最终仅剩下5个例外族；在每个亏格下，所有其他可容许特征都是可实现的。这5个族中有2个似乎是新的，且均存在于亏格2中；其中4个族超出了gcd条件的范围，该条件留下的全部空白区域在此得到解决。非空性部分通过构造性方法实现：在每个亏格下构造一个单顶点方砖曲面，以及一种局部手术，该手术将一个锥拆分为两个具有规定角度的锥，同时保持亏格、其他锥以及im ρ不变。随后有两个扩展：带边界的曲面、特征对齐设置，以及与固定共形结构下Abel-Jacobi准则的关系。
英文摘要
A seamless parametrization of a closed oriented surface carries a discrete invariant, its holonomy signature: the cone angles, all multiples of $π/2$, together with the rotational holonomy $ρ\colon H_1(M\setminus C)\to\mathbb{Z}_4$ of the induced cross field. This is the datum a quadrangulation prescribes, and it decides whether any parametrization exists at all. Shen, Zhu, Capouellez, Panozzo, Campen and Zorin asked which signatures occur and gave a sufficient condition of gcd type; which signatures are realizable has remained open. We answer the question. A Reduction Lemma shows that the mapping class group acts on signatures with fixed cone angles with orbits classified by the subgroup $\mathrm{im}\,ρ\le\mathbb{Z}_4$ alone, so at most three cases survive per angle multiset instead of $4^{2g}$. A dictionary then identifies seamless parametrizations with meromorphic 4-differentials, under which $\mathrm{im}\,ρ$ measures primitivity, and realizability becomes non-emptiness of a stratum of primitive $k$-differentials with $k=4/d$ and $\mathrm{im}\,ρ=\langle d\rangle$. Unwinding this against the known classification of such strata leaves exactly five exceptional families; every other admissible signature is realizable, in every genus. Two of the five appear to be new, and both live in genus two. Four of the five lie outside the gcd condition, and the whole region it leaves open is settled here. The non-emptiness half is made constructive by an explicit one-vertex square-tiled surface in every genus together with a local surgery that splits one cone into two of prescribed angles, leaving the genus, the other cones and $\mathrm{im}\,ρ$ untouched. Two extensions follow: surfaces with boundary, the feature-aligned setting, and the relation to the Abel-Jacobi criterion at a fixed conformal structure.
