Which Holonomy Signatures Are Realizable? A Complete Answer for Closed Surfaces

Leyi Zhu

Which Holonomy Signatures Are Realizable? A Complete Answer for Closed Surfaces

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Pith's one-line read The paper proves a complete classification of which holonomy signatures—cone angles plus rotational holonomy—are realizable by seamless parametrizations of closed surfaces: every Gauss–Bonnet-admissible signature except five explicit famili

desk verdict A real answer to a real open question, with a clean reduction and a five-family classification, but the completeness of the main theorem is explicitly conditional on the external Gendron–Tahar classification. read the letter →

arxiv 2608.01444 v1 pith:A4QQXTJG submitted 2026-08-02 math.GT

Leyi Zhu This is my paper · ORCID

classification math.GT MSC 57K2030F3032G15

keywords holonomysignatureseamlessparametrizationquadrangulationmeromorphic4-differentialsprimitivek-differentialsmodulistratamappingclassgroupflatconemetrics

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper settles, for closed oriented surfaces, which holonomy signatures are realizable: which cone-angle data plus rotational holonomy can actually be carried by a seamless parametrization, equivalently by a quadrangulation. The answer is that every Gauss-Bonnet-admissible signature is realizable except five explicit families: on the torus, no cones with non-zero holonomy, cones $3\pi/2$ and $5\pi/2$ with any holonomy, and cones $\pi$ and $3\pi$ with holonomy $2\mathbb{Z}_4$; on genus two, a single $6\pi$ cone or a $3\pi/5\pi$ pair with holonomy $2\mathbb{Z}_4$. The proof shows that realizability is the same as non-emptiness of a stratum of primitive $k$-differentials, where the holonomy subgroup $\operatorname{im}\rho$ determines $k=4/d$, then reads off the known classification of empty strata. A separate construction produces explicit minimal square-tiled witnesses for most signatures. This matters because a signature is exactly the data a quadrangulation prescribes, so the result tells practitioners whether a parametrization exists before trying to build one.

What carries the argument

The load-bearing machinery is the dictionary pairing seamless parametrizations with flat $\mathbb{Z}_4$ cone metrics and then with meromorphic 4-differentials: a signature with $\operatorname{im}\rho=d\mathbb{Z}_4$ is realizable iff the stratum of primitive $(4/d)$-differentials with orders $m_i/d$ is non-empty (Theorem 3.6). The Reduction Lemma (Theorem 3.4) supplies the input that for fixed cone angles and genus $g\ge 1$, two signatures are mapping-class equivalent iff their $\operatorname{im}\rho$ agree, cutting the problem to three cases per angle multiset. Lemma 2.6 is the identity that primitivity is measured by $\operatorname{im}\rho$: $q=\eta^d$ exactly when $\operatorname{im}\rho\su

What would settle it

Find a genus $g\ge 1$ profile $\mu$ of a primitive $k$-differential that satisfies the Gauss-Bonnet degree and pole bounds but is not one of $(1,-1)$, the empty genus-one profiles, or $(4),(3,1)$ for $k=2$, and whose stratum is empty. By the paper's own dictionary that would yield a sixth unrealizable holonomy signature and refute Theorem 3.10. A more targeted check: verify that no genus-two quad mesh with one $6\pi$ cone and $\operatorname{im}\rho=2\mathbb{Z}_4$ exists at any size—the paper claims none because the reduced stratum $Q(4)$ is empty.

Watch

Extended reading notes

Core claim

The central discovery is Theorem 3.10: for a closed oriented surface of any genus, a holonomy signature $(g,m,\rho)$ is realizable if and only if it is not one of five families: (1) torus with no cone points and $\operatorname{im}\rho\neq 0$; (2) torus with cones of angles $3\pi/2$ and $5\pi/2$; (3) torus with cones $\pi$ and $3\pi$ and $\operatorname{im}\rho=2\mathbb{Z}_4$; (4) genus-two surface with one cone of angle $6\pi$ and $\operatorname{im}\rho=2\mathbb{Z}_4$; (5) genus-two surface with cones $3\pi$ and $5\pi$ and $\operatorname{im}\rho=2\mathbb{Z}_4$. Two mechanisms carry the proof. First, the Reduction Lemma (Theorem 3.4) shows that for fixed cone angles the mapping class group orb

Load-bearing premise

The whole five-family answer for genus $g\ge 1$ rests on the external classification of empty primitive $k$-differential strata being complete; if that classification has missed any empty stratum, the paper's list is missing a corresponding unrealizable signature. The paper itself says it takes the completeness of that list 'purely on trust.'

Editorial extensions

If this is right

The gcd condition of the prior sufficient criterion is not necessary: the entire region where the gcd of the cone orders is not $1$ is now settled, with exactly four unrealizable signatures there.
If some cone angle is an odd multiple of $\pi/2$, the holonomy $\rho$ is irrelevant: all signatures with those cone angles form a single mapping class group orbit, so realizability depends only on the cone angles.
A genus-two surface whose cross field splits globally into two line fields (holonomy $2\mathbb{Z}_4$) cannot be quadrangulated with a single $6\pi$ cone or with a $3\pi/5\pi$ pair.
Every other Gauss-Bonnet-admissible signature is realizable in every genus, and the positive half is witnessed by explicit one-vertex square-tiled surfaces together with a local surgery that splits one cone into two prescribed cones, often attaining the minimal square count $2g-2+n$.
On surfaces with boundary, the Reduction Lemma holds with boundary turnings included in the subgroup, so a boundary component carrying an odd number of odd-angle corners forces $\mathbb{Z}_4$ holonomy and makes the holonomy along homology loops irrelevant.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Because realizability depends only on the subgroup $\operatorname{im}\rho$, a practical pre-check for quad meshing could decide existence from cone angles plus whether the cross field is globally a vector field, a pair of line fields, or a genuine four-prong field—without knowing the full holonomy character.
If the termination gap in the constructive algorithm were closed—if every cone of valence $w$ on a genus $\ge 2$ mesh always carries a loop of the needed gap—the classification would become fully constructive and independent of the cited stratum classification.
The boundary doubling discussion suggests that the real-stratum analogue, $k$-differentials invariant under a prescribed anti-holomorphic involution, is likely non-empty whenever Gauss-Bonnet and the parity constraint allow, which would make feature-aligned parametrization existence unconditional.
At a fixed conformal structure, holonomy is not free but Abel-Jacobi computed; the paper's topological answer implies that failures there come from the defined locus being empty, not from the pointwise criterion itself.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The short version: this paper answers a long-standing question and mostly earns its claim, but the completeness of the headline theorem is only as solid as the external classification it leans on, and the author says so plainly. That is the one thing to keep in mind.

What is genuinely new is the structure. The Reduction Lemma collapses the mapping class group orbits to im(rho) alone, so at most three signatures survive per angle multiset instead of 4^(2g). The dictionary to primitive k-differententials turns realizability into a stratum non-emptiness question, and unwinding that gives the five unrealizable families. The genus-one cases are reproved independently via Abel–Jacobi, and the explicit square-tiled construction with the certificate theorem is a nice bonus: outputs are self-certifying gluings whose genus, cone orders, and im(rho) can be read off the dart pairing. I also credit the paper for saying what it does not prove: §2.3 takes (S7) on trust, and §6.3 lists the open gaps.

The soft spots are real but mostly peripheral. The load-bearing one is (S7): every non-emptiness statement for g≥1 passes through the Gendron–Tahar list, and if that list is incomplete, Theorem 3.10 is false. The paper does not verify it; the stress test is accurate about that. I do not think this is a fatal flaw—external theorems get cited all the time—but it means the central claim is conditional, and a referee should push on it. The other gaps are disclosed and minor by comparison: the constructive iteration lacks a termination proof, the abstract overstates by suggesting a theorem for all genera when §4.3 says no such theorem is proved, d=2 base meshes exist only up to genus 8, and the figure-eight move has no general lemma.

The citation pattern is fine. [14] is the author's own prior work, but it is used as the source of the question and of a sufficient condition, not as black-box support for the new claims. The central argument is coherent, and the honest qualifiers are where I would put them.

Bottom line: if you work on seamless parametrizations, flat cone metrics, or k-differential strata, this is worth a careful read. It deserves a serious referee—ideally someone who knows the Gendron–Tahar classification from the inside. I would send it out and ask the author to address the S7 dependence explicitly, not reject it.

Referee Report

0 major / 6 minor

Summary. The paper gives a complete classification of realizable holonomy signatures on closed oriented surfaces. A holonomy signature records cone angles (multiples of π/2) and the rotational holonomy ρ: H1(M\C) → Z4 of a seamless parametrization. The main theorem (Theorem 3.10) states that a Gauss–Bonnet admissible signature is realizable unless it belongs to one of five explicit families in Table 1: three torus families (no cones with im ρ ≠ 0; a 3π/2 and 5π/2 pair; a π and 3π pair with im ρ = 2Z4) and two genus-two families (a single 6π cone with im ρ = 2Z4; a 3π and 5π pair with im ρ = 2Z4). The proof proceeds via a dictionary between seamless parametrizations and meromorphic 4-differentials (Lemmas 2.3, 2.5, 2.6), a Reduction Lemma showing that mapping-class orbits are classified by im ρ (Theorem 3.4), and a reduction of realizability to non-emptiness of primitive k-differential strata (Theorem 3.6). Genus 0 is handled by Troyanov's theorem, genus 1 by an Abel–Jacobi argument, and genus ≥ 1 by the external Gendron–Tahar classification quoted as (S7). Section 4 develops explicit square-tiled witnesses and a certificate theorem for output meshes; Section 5 analyzes the region outside the previous gcd sufficient condition; Section 6 treats boundary/feature-aligned surfaces and fixed conformal structures.

Significance. If correct, this settles a question that had been open since the sufficient gcd-type condition of Shen–Zhu–Capouellez–Panozzo–Campen–Zorin. The structural insight is strong: for fixed cone angles, only the subgroup im ρ ≤ Z4 matters, reducing what naively looks like 4^{2g} cases to at most three. The dictionary with primitive k-differentials is clean and explains the known torus exception as an empty stratum. Two of the five exceptional families (the genus-two entries) appear to be new and are plausible and interesting. The paper is unusually transparent: it states exactly which external theorem supplies non-emptiness for g ≥ 1, explicitly flags what is not proved (termination of Algorithm 4.5, the d = 2 base meshes beyond genus 8, the figure-eight move lemma), and frames these as open problems. The constructive part provides explicit minimal square-tiled witnesses whenever the algorithm returns, with a certificate theorem that makes each output self-verifying. These are genuine strengths.

minor comments (6)

[Theorem 3.10, §2.3] The completeness of the classification for g ≥ 1 is exactly the completeness of the quoted external list (S7). The paper acknowledges this in §2.3, but the title and abstract say 'complete answer' without immediate qualification. Please add a sentence at the statement of Theorem 3.10 making explicit that, for g ≥ 1, the non-emptiness direction is contingent on the Gendron–Tahar classification (S7), and perhaps restate (S7) as a displayed theorem rather than as a black box in text. This is not an objection to using a published theorem, but it is the single most load-bearing input and should be unmissable where the theorem is stated.
[Abstract, §4, Remark 4.7] The abstract says the non-emptiness half is 'made constructive' by an explicit one-vertex square-tiled surface in every genus plus a local surgery. This overstates what is proved: the base mesh of Lemma 4.3 covers d = 1 and d = 4 in every genus, but for d = 2 bases are only recorded for 2 ≤ g ≤ 8, and Algorithm 4.5 has no termination proof, as Remark 4.7 itself clearly states. Suggest a wording such as 'made constructive on a finite range, with a certificate theorem for every output' or similar.
[Algorithm 4.5, §4.3] The d = 2 base meshes for 2 ≤ g ≤ 8 are said to be 'recorded' but are not displayed or listed in the paper. Since Theorem 4.6 makes each output a certificate, it would be helpful to include the gluing arrays in an appendix or ancillary file so a reader can reproduce the search and verify the claims, especially given that the construction is one of the paper's advertised contributions.
[§6.1, Corollary 6.6] The sentence 'Over all 3058 admissible feature-aligned signatures whose corner count fits in four squares, Corollary 6.6 never applies' appears to be a computational claim. Please provide the enumeration data, the code, or a reproducible description of how this number was obtained. As written, it is not checkable from the paper.
[Notation, Lemma 6.5] The notation 'imeρ' for the holonomy of the double is hard to read and not defined formally. Please introduce a symbol such as \widetilde{\rho} and state its domain explicitly. Also, in the displayed formula after Lemma 6.5, the inclusion 'imeρ ⊇ ⟨imρ, 2a mod 4⟩' deserves a short explanation of why the new loops contribute 2a mod 4.
[Lemma 3.2, Step 1] The letter D is used for the gcd of a lifted vector in Lemma 3.2 and later for the subgroup generated by the cone orders. This is a minor clash; consider using G or c for the gcd. The proof itself is sound, but the notation makes the argument slightly harder to follow.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Theorem 3.10 is a proved reduction to external classifications (S6)/(S7), with genus 1 reproved internally; acknowledged external dependence is a limitation, not circularity.

full rationale

The paper's derivation chain is non-circular. The Reduction Lemma (Theorem 3.4) and the stratum dictionary (Theorem 3.6) are proved internally from point pushing, symplectic transitivity, and the flat-cone/4-differential equivalences (Lemmas 2.3, 2.5, 2.6); they do not assume the target classification. The central completeness result, Theorem 3.10, is obtained by unwinding Theorem 3.6 against the external Gendron–Tahar classification (S7), quoted in §2.3, plus Troyanov (S6) for genus 0; the genus-one cases are reproved independently in Proposition 3.8. The one self-citation, [14] (which includes the present author), is used only as the source of the question and prior sufficient condition, and is explicitly contrasted with the new result; no step's truth depends on it. The paper explicitly flags its limitations: it 'take[s] purely on trust that the list stops there' (§2.3), lists removing the (S7) dependence as open problem §6.3(1), and admits in Remark 4.7 that Algorithm 4.5 has no termination proof and that d=2 base meshes are available only for 2≤g≤8. These are external-dependence and incompleteness caveats, not circular reductions: (S7) is a prior theorem about k-differential strata whose statement does not include holonomy signatures and is not derived from this paper's conclusions. No fitted parameter is relabeled as a prediction, no uniqueness theorem is imported from the authors' own prior work, and the dictionary is a proved equivalence rather than a renaming of a known result.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters or fitted constants appear; the paper is a pure classification theorem. The central claim rests on two external existence classifications (Troyanov for the sphere, Gendron-Tahar for all higher genera), plus routine topological facts. No new entities such as particles, forces, or dimensions are postulated; the holonomy signature concept is inherited from [14].

assumptions (5)

standard math Existence of geodesic triangulations and common refinements (S1,S2).
Used in Lemma 2.4 to promote flat cone metrics to piecewise-linear seamless atlases on a given triangulation; standard surface topology.
standard math Symplectic transitivity and reduction: Sp(2g,Z) acts transitively on primitive vectors and maps onto Sp(2g,Z/N) (S3,S4).
Used in Lemma 3.2 and Theorem 3.4 to classify orbits of holonomy vectors by gcd content; classical.
standard math The mapping class group of a genus-g surface with one boundary component maps onto Sp(2g,Z) (S5).
Used in Lemma 3.3 to realize all symplectic automorphisms of H1(M;Z4) while fixing a base signature.
domain assumption Troyanov's theorem: on the sphere, any n≥3 positive cone angles satisfying Gauss-Bonnet are realizable by a flat cone metric (S6).
Sole input for genus zero in Proposition 3.7; not reproved, standard external theorem.
domain assumption Gendron-Tahar classification (S7): for g≥1, primitive k-differential strata are empty exactly for g=1, mu=(1,-1); g=1, mu=empty, k≥2; and g=2, k=2, mu=(4) or (3,1).
The load-bearing external input for all g≥1 non-emptiness in Theorem 3.10; the author explicitly does not prove the list is complete.

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Cite this review

Pith. "Pith review of Which Holonomy Signatures Are Realizable? A Complete Answer for Closed Surfaces." pith.science (2026). https://pith.science/paper/A4QQXTJG

@misc{pith2026260801444,
  author       = {Pith},
  title        = {Pith review of: Which Holonomy Signatures Are Realizable? A Complete Answer for Closed Surfaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/A4QQXTJG}},
  note         = {Machine review of arXiv:2608.01444}
}

abstract

A seamless parametrization of a closed oriented surface carries a discrete invariant, its holonomy signature: the cone angles, all multiples of $\pi/2$, together with the rotational holonomy $\rho\colon H_1(M\setminus C)\to\mathbb{Z}_4$ 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 $\mathrm{im}\,\rho\le\mathbb{Z}_4$ 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 $\mathrm{im}\,\rho$ measures primitivity, and realizability becomes non-emptiness of a stratum of primitive $k$-differentials with $k=4/d$ and $\mathrm{im}\,\rho=\langle d\rangle$. 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 $\mathrm{im}\,\rho$ untouched. Two extensions follow: surfaces with boundary, the feature-aligned setting, and the relation to the Abel-Jacobi criterion at a fixed conformal structure.

Figures

Figures reproduced from arXiv: 2608.01444 by the authors.

**Figure 1.** The three possible values of d, the generator of im ρ, and what each means for the cross field. When im ρ = 0 the cross lifts to a global vector field. When im ρ = 2Z4 it does not, but it still splits globally into two distinguishable line fields, drawn solid and dashed. When im ρ = Z4 transport around some loop cyclically permutes the four prongs and no such splitting exists. By the Reduction Lemma these three case… view at source ↗

**Figure 2.** The correspondences of §2.4. Reading left to right, a seamless parametrization is a flat cone metric with holonomy in Z4 (Lemma 2.3), and such a metric is |q| 1/2 for a meromorphic 4-differential q whose zeros and poles are the cones (Lemma 2.5). Under this correspondence im ρ measures primitivity: q is a d-th power exactly when im ρ ⊆ dZ4, so im ρ = dZ4 says that q = η d with η primitive (Lemma 2.6). Combined with … view at source ↗

**Figure 3.** Point pushing (Lemma 3.1). Dragging a cone of order [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: The five unrealizable holonomy signatures of Table 1; dots mark cones. Families 1 to [PITH_FULL_IMAGE:figures/full_fig_p014_4.png]

**Figure 5.** Figure 5: The splitting surgery of Lemma 4.1, seen in the link of the cone. (a) The [PITH_FULL_IMAGE:figures/full_fig_p017_5.png]

**Figure 6.** Figure 6: The base mesh of Lemma 4.3 at g = 2: N = 2g−1 = 3 unit squares, side s of square f carrying dart 4f + s, numbered counterclockwise from the bottom. The six gluings, one colour each, pair every dart once, so the mesh is closed. Here ν = α ◦ σ −1 is a single 12-cycle, so…

**Figure 7.** Figure 7: Two explicit minimal witnesses on the torus, in the notation of Figure 6: side [PITH_FULL_IMAGE:figures/full_fig_p023_7.png]

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

15 extracted references · 9 canonical work pages

[14]

H. Shen, L. Zhu, R. Capouellez, D. Panozzo, M. Campen, D. Zorin,Which cross fields can be quadrangulated? Global parameterization from prescribed holonomy signatures, ACM Trans. Graph. 41(2022), no. 4, Art. 59;doi:10.1145/3528223.3530187

arXiv 2022
[1]

Campen, H

M. Campen, H. Shen, J. Zhou, D. Zorin,Seamless parametrization with arbitrary cones for arbitrary genus, ACM Trans. Graph.39(2019), no. 1, Art. 2, 19 pp.;doi:10.1145/3360511

doi:10.1145/3360511 2019
[2]

Seamless Parametrization in Penner Coordinates

R. Capouellez, D. Zorin,Seamless parametrization in Penner coordinates, ACM Trans. Graph.43 (2024), no. 4, Art. 61;doi:10.1145/3658202;arXiv:2407.21342

work page Pith review arXiv 2024
[3]

Capouellez, R

R. Capouellez, R. Singh, M. Heistermann, D. Bommes, D. Zorin,Feature-aligned parametrization in Penner coordinates, ACM Trans. Graph.44(2025), no. 4;doi:10.1145/3731216

doi:10.1145/3731216 2025
[4]

Abelian differentials with prescribed singularities

Q. Gendron, G. Tahar,Diff´ erentielles ab´ eliennes ` a singularit´ es prescrites[Abelian differentials with prescribed singularities], J. ´Ec. polytech. Math.8(2021), 1397–1428;doi:10.5802/jep.174; arXiv:2103.03165

work page Pith review arXiv 2021
[5]

Quadratic differentials with prescribed singularities

Q. Gendron, G. Tahar,Diff´ erentielles quadratiques ` a singularit´ es prescrites[Quadratic differentials with prescribed singularities], Canad. J. Math.77(2025), no. 5, 1762–1826;arXiv:2111.12653

work page Pith review arXiv 2025
[6]

Diff\'erentielles \`a singularit\'es prescrites

Q. Gendron, G. Tahar,Diff´ erentielles ` a singularit´ es prescrites,arXiv:1705.03240v2(2020); The- orem 1.4 is our (S7). Thek-differential part has since appeared ask-diff´ erentielles ` a singularit´ es prescrites, Ann. Fac. Sci. Toulouse Math. (6)34(2025), no. 4, 901–1006. 27

work page Pith review arXiv 2020
[7]

There is no triangulation of the torus with vertex degrees 5, 6, ..., 6, 7 and related results: Geometric proofs for combinatorial theorems

I. Izmestiev, R. B. Kusner, G. Rote, B. Springborn, J. M. Sullivan,There is no triangulation of the torus with vertex degrees5,6, . . . ,6,7and related results: geometric proofs for combinatorial theo- rems, Geom. Dedicata166(2013), 15–29;doi:10.1007/s10711-012-9782-5;arXiv:1207.3605

work page Pith review arXiv 2013

Show all 15 references

[8]

Kontsevich, A

M. Kontsevich, A. Zorich,Connected components of the moduli spaces of Abelian differentials with prescribed singularities, Invent. Math.153(2003), no. 3, 631–678; doi:10.1007/s00222-003-0303-x

2003 doi
[9]

Lanneau,Connected components of the strata of the moduli spaces of quadratic differentials, Ann

E. Lanneau,Connected components of the strata of the moduli spaces of quadratic differentials, Ann. Sci. ´Ec. Norm. Sup´ er. (4)41(2008), no. 1, 1–56;doi:10.24033/asens.2062

2008 doi
[10]

N. Lei, X. Zheng, J. Jiang, Y.-Y. Lin, D. X. Gu,Quadrilateral and hexahedral mesh generation based on surface foliation theory, Comput. Methods Appl. Mech. Engrg.316(2017), 758–781; doi:10.1016/j.cma.2016.09.044

2017 doi
[11]

N. Lei, X. Zheng, Z. Luo, F. Luo, X. Gu,Quadrilateral mesh generation II: meromorphic quartic differentials and Abel–Jacobi condition, Comput. Methods Appl. Mech. Engrg.366(2020), 112980; doi:10.1016/j.cma.2020.112980;arXiv:1907.00216

2020
[12]

Zheng, Y

X. Zheng, Y. Zhu, N. Lei, Z. Luo, X. Gu,Quadrilateral mesh generation III: optimizing singularity configuration based on Abel–Jacobi theory, Comput. Methods Appl. Mech. Engrg.387(2021), 114146; doi:10.1016/j.cma.2021.114146;arXiv:2007.07334

2021
[13]

Masur, J

H. Masur, J. Smillie,Quadratic differentials with prescribed singularities and pseudo-Anosov diffeo- morphisms, Comment. Math. Helv.68(1993), no. 2, 289–307;doi:10.1007/BF02565820

1993 doi
[15]

Troyanov,Prescribing curvature on compact surfaces with conical singularities, Trans

M. Troyanov,Prescribing curvature on compact surfaces with conical singularities, Trans. Amer. Math. Soc.324(1991), no. 2, 793–821;doi:10.1090/S0002-9947-1991-1005085-9. 28

1991 doi

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [14]

H. Shen, L. Zhu, R. Capouellez, D. Panozzo, M. Campen, D. Zorin,Which cross fields can be quadrangulated? Global parameterization from prescribed holonomy signatures, ACM Trans. Graph. 41(2022), no. 4, Art. 59;doi:10.1145/3528223.3530187

arXiv 2022

[2] [1]

Campen, H

M. Campen, H. Shen, J. Zhou, D. Zorin,Seamless parametrization with arbitrary cones for arbitrary genus, ACM Trans. Graph.39(2019), no. 1, Art. 2, 19 pp.;doi:10.1145/3360511

doi:10.1145/3360511 2019

[3] [2]

Seamless Parametrization in Penner Coordinates

R. Capouellez, D. Zorin,Seamless parametrization in Penner coordinates, ACM Trans. Graph.43 (2024), no. 4, Art. 61;doi:10.1145/3658202;arXiv:2407.21342

work page Pith review arXiv 2024

[4] [3]

Capouellez, R

R. Capouellez, R. Singh, M. Heistermann, D. Bommes, D. Zorin,Feature-aligned parametrization in Penner coordinates, ACM Trans. Graph.44(2025), no. 4;doi:10.1145/3731216

doi:10.1145/3731216 2025

[5] [4]

Abelian differentials with prescribed singularities

Q. Gendron, G. Tahar,Diff´ erentielles ab´ eliennes ` a singularit´ es prescrites[Abelian differentials with prescribed singularities], J. ´Ec. polytech. Math.8(2021), 1397–1428;doi:10.5802/jep.174; arXiv:2103.03165

work page Pith review arXiv 2021

[6] [5]

Quadratic differentials with prescribed singularities

Q. Gendron, G. Tahar,Diff´ erentielles quadratiques ` a singularit´ es prescrites[Quadratic differentials with prescribed singularities], Canad. J. Math.77(2025), no. 5, 1762–1826;arXiv:2111.12653

work page Pith review arXiv 2025

[7] [6]

Diff\'erentielles \`a singularit\'es prescrites

Q. Gendron, G. Tahar,Diff´ erentielles ` a singularit´ es prescrites,arXiv:1705.03240v2(2020); The- orem 1.4 is our (S7). Thek-differential part has since appeared ask-diff´ erentielles ` a singularit´ es prescrites, Ann. Fac. Sci. Toulouse Math. (6)34(2025), no. 4, 901–1006. 27

work page Pith review arXiv 2020

[8] [7]

There is no triangulation of the torus with vertex degrees 5, 6, ..., 6, 7 and related results: Geometric proofs for combinatorial theorems

I. Izmestiev, R. B. Kusner, G. Rote, B. Springborn, J. M. Sullivan,There is no triangulation of the torus with vertex degrees5,6, . . . ,6,7and related results: geometric proofs for combinatorial theo- rems, Geom. Dedicata166(2013), 15–29;doi:10.1007/s10711-012-9782-5;arXiv:1207.3605

work page Pith review arXiv 2013