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Shortest filling geodesics on hyperbolic surfaces

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Pith's one-line read For every closed orientable hyperbolic surface of genus g, the shortest total length of a filling multi-geodesic is exactly half the perimeter of the regular right-angled hyperbolic (8g−4)-gon, and a single filling geodesic attains it.

desk verdict Likely-true main theorem, but the proof has a load-bearing gap in the isoperimetric inequality and a concrete miscount in the construction; worth refereeing after revision. read the letter →

arxiv 2506.12465 v1 pith:NKKJTKL5 submitted 2025-06-14 math.GT

Yue Gao , Jiajun Wang , Zhongzi Wang This is my paper · ORCID
classification math.GT MSC 57K2053C2251M10
keywords fillinggeodesicshyperbolicsurfacesisoperimetricinequalityright-angledpolygonperimetergeodesicgraphmodulispacekissingnumber
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

This paper answers a natural question in hyperbolic geometry: among all genus-g hyperbolic surfaces, what is the minimal total length of a set of closed geodesics that cuts the surface into disks? The answer is exactly half the perimeter of the regular right-angled hyperbolic (8g−4)-gon, and the minimum is realized by a single filling geodesic. The proof combines a reduction that eliminates triangles from the complementary regions with a new isoperimetric inequality for hyperbolic polygons.

What carries the argument

The argument rests on two mechanisms: a reduction (Theorem 2.1) that converts any filling multi-geodesic into a filling geodesic graph of no greater length whose complementary polygons each have at least five sides, with the side-count identity sum(m_i − 4) = 8g − 8; and an isoperimetric inequality (Theorem 3.1) that bounds the total perimeter of such a polygon family below by P_{8g−4}. The inequality is proved by studying perimeter functions of regular hyperbolic polygons and relies on a generalization of Sanki-Vadnere's even-polygon isoperimetric result.

What would settle it

Construct a genus-g hyperbolic surface and a filling multi-geodesic of total length strictly less than 1/2 P_{8g−4}, or exhibit a filling multi-geodesic of length exactly 1/2 P_{8g−4} whose complement is not a regular right-angled (8g−4)-gon.

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Extended reading notes

Core claim

The central discovery is that the filling length infimum over the moduli space of genus-g hyperbolic surfaces equals 1/2 P_{8g−4}, where P_k is the perimeter of the regular right-angled hyperbolic k-gon. The proof proceeds in two steps: first, any filling multi-geodesic is shortened to a filling geodesic graph whose complement has only polygons with at least five sides and satisfies sum(m_i − 4) = 8g − 8; second, an isoperimetric inequality shows the total perimeter of such a collection is at least P_{8g−4}, with equality only for a single regular right-angled (8g−4)-gon. An explicit construction exhibits a single filling geodesic of this length, so the bound is sharp.

Load-bearing premise

The lower bound rests on the reduction in Theorem 2.1, which asserts that every filling multi-geodesic can be replaced by a filling geodesic graph of no greater length whose complementary polygons all have at least five sides; if that reduction produced triangles or increased length, the isoperimetric inequality of Theorem 3.1 could not be applied and the bound would not follow.

Editorial extensions

If this is right

  • The bound is sharp for every genus g ≥ 2: a single filling geodesic whose complement is a regular right-angled (8g−4)-gon has length exactly 1/2 P_{8g−4}.
  • Equality is rigid: a filling multi-geodesic of length 1/2 P_{8g−4} must be a single geodesic whose complement is the regular right-angled (8g−4)-gon.
  • For surfaces whose systole set fills, the kissing number satisfies sys(X) · kiss(X) ≥ 1/2 P_{8g−4}, yielding kiss(X) ≥ (3.525) g / log g for large g, improving the previous constant.
  • The reduction and isoperimetric inequality apply to arbitrary filling multi-geodesics, including those with multiple intersection points and self-intersections, settling a conjecture previously known only for filling pairs.

Reading between the lines

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

  • The triangle-free condition in the reduction is essential, since the isoperimetric inequality fails for triangles; this suggests that any relaxation of the filling-graph hypothesis would need an explicit correction term that accounts for triangular regions.
  • The same two-step strategy (reduce the graph, then apply a polygon isoperimetric inequality) could be adapted to other settings, such as filling curves on punctured surfaces or on non-orientable surfaces, where the relevant regular polygon side count would change.
  • The equality case forces the surface to be a regular polygonal gluing, hinting that global minimizers of filling length in moduli space are highly symmetric; it would be natural to ask whether these are the only critical points of the filling-length function.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 3 minor

Summary. The paper claims to determine the minimal length of a filling multi-geodesic on a closed orientable hyperbolic surface of genus g, minimized over moduli space. Theorem 1.1 states this minimum equals half the perimeter of a regular right-angled (8g−4)-gon, and that it is realized by a single filling geodesic. The proof has two main ingredients: Theorem 2.1, which replaces a filling multi-geodesic by a filling geodesic graph of no larger length whose complementary components are polygons with at least five sides and satisfy a side-count identity, and Theorem 3.1, an isoperimetric inequality comparing the perimeter sum of such polygons with the perimeter of a regular polygon of the same area. Section 4 constructs a hyperbolic surface and a single filling geodesic whose complement is the desired regular polygon.

Significance. If correct, the result settles a natural question about filling geodesics and extends the Aougab–Huang conjecture from filling pairs to arbitrary filling multi-geodesics. The construction of a single filling geodesic attaining the bound is explicit, and the corollaries on the kissing number of surfaces with filling systole are of independent interest. The paper also provides a clean statement of the isoperimetric inequality for polygons with at least four sides, which is a useful contribution in itself. However, the proof of the isoperimetric theorem is currently incomplete, so the main result is not yet established as written.

major comments (2)
  1. [Section 2.3, proof of Theorem 2.1] The theorem is stated for arbitrary hyperbolic mi-gons, but the proof from Proposition 3.8 onward only treats regular polygons. Proposition 3.8 assumes D1 and D2 are regular hyperbolic polygons, Lemma 3.10 assumes each Di is regular, and the induction in the proof of Theorem 3.1 compares the regular replacements eD_j. There is no argument showing that, among hyperbolic polygons with a fixed number of sides and fixed area, the regular polygon minimizes perimeter. Reference [10] (Bezdek) is listed but never cited in the proof. Consequently the key step in the proof of Theorem 1.1, namely ℓ(G) = 1/2 Σ Perim(D_i) ≥ 1/2 P_{8g−4}, does not follow for the arbitrary polygons produced by Theorem 2.1. This is the load-bearing connection between the graph reduction and the numerical lower bound, so the proof of Theorem 3.1 is incomplete as written.
  2. [Section 2.3, proof of Theorem 2.1] The iterative construction ends with the assertion: 'Since Γ is a finite filling graph, the repeating process will eventually stop.' This is not justified in the general case. The replacement step replaces an arc β by a shortest proper geodesic arc β′ that may not lie in Γ, so the intermediate graph Θ_i is no longer a subgraph of Γ. Finiteness of Γ therefore does not, by itself, bound the number of iterations. A rigorous termination argument—for instance, a monotone complexity measure for the pair (Θ_i, G_i) or a bound on the number of essential cutting curves needed to fill the surface—is required. The termination of the process is essential to the existence of the graph G, so this gap also affects the lower bound.
minor comments (3)
  1. [References] Reference [10] (Bezdek) is never cited in the body of the paper. Either it should be used to supply the missing regularization step in Theorem 3.1, or it should be removed.
  2. [Section 4] The claim that the curve α in Figure 12 is filling and that its complement is an (8g−4)-gon is stated without proof. Since the construction of the extremal surface depends on this assertion, a brief justification or a more detailed reading of the figure would be helpful.
  3. [Throughout] There are several typographical errors, including 'SHOR TEST' in the title, 'olygons' in Section 2.2, 'essencity' in Section 2.3, and some malformed summation indices such as 'kX k=1'. The manuscript would benefit from a careful proofreading pass.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the main lower bound is derived from an independent isoperimetric inequality and a self-contained graph-reduction lemma; the noted proof gap in Theorem 3.1 is a completeness issue, not circularity.

full rationale

The derivation chain is not circular. Theorem 2.1 is a self-contained combinatorial-geometric reduction: it builds a filling convex geodesic graph G from Gamma with length(G) <= length(Gamma) and side counts m_i >= 5 satisfying sum(m_i - 4) = 8g - 8, using only Gauss-Bonnet and geodesic convexity arguments (Section 2). The lower bound then rests on Theorem 3.1, whose proof uses Proposition 3.5 from Sanki-Vadnere [29] - prior independent published work not by these authors - and calculus on the regular hyperbolic polygon perimeter function P_n(a). No equation in the paper defines the target (1/2)P_{8g-4} in terms of the quantities being bounded, and no fitted parameter is renamed as a prediction. The paper's self-citations ([32], [38]) appear only in background remarks on non-simple geodesics and are not load-bearing. The legitimate concern in this manuscript is a completeness gap in the proof of Theorem 3.1: the statement allows arbitrary hyperbolic m_i-gons, but from Proposition 3.8 onward the proof silently treats the D_i as regular (e.g., 'We consider a series of hyperbolic regular polygons eD1, eD2, ..., eDk' and then applies Proposition 3.8 to D1,D2), and the cited isoperimetric result [10] is never invoked to justify this regularization. That is a missing argument or correctness issue, not a circular reduction: Theorem 3.1 is not assumed in its own proof, and the main theorem would follow if the regularization lemma were supplied.

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

No free parameters were fitted and no new entities were introduced. The argument rests on standard background theorems (Gauss-Bonnet, polygon isoperimetry, Sanki-Vadnere's perimeter estimates) and on the graph-reduction lemma of Section 2. The only paper-specific unproved input is the topological construction in Figure 12, where the double-point count needs correction.

assumptions (5)
  • standard math Gauss-Bonnet theorem for hyperbolic surfaces and polygons
    Used throughout to compute areas and to rule out impossible configurations, for example in Lemma 2.3 and throughout Section 3.
  • standard math Hyperbolic polygon isoperimetric inequality: among convex hyperbolic m-gons of fixed area, the regular m-gon has minimal perimeter (Bezdek [10])
    Needed in Theorem 3.1 to replace each polygon D_i by a regular polygon eD_i with the same area; the proof does not explicitly state this reduction.
  • domain assumption Sanki-Vadnere's Proposition 3.5 (convexity and monotonicity properties of perimeter functions)
    Imported from [29] and used as a black box in Lemma 3.7 and Proposition 3.6.
  • standard math Existence of a minimizer via Mumford compactness and the collar lemma
    Used in the introduction to justify that Question 1 has a well-defined minimum.
  • ad hoc to paper Topological assertion that the curve in Figure 12 is filling with complement an (8g-4)-gon
    Section 4 asserts this from the figure; the stated double-point count contains an apparent typo (2g-3 versus 2g-1), so the construction should be checked for correctness.

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Pith. "Pith review of Shortest filling geodesics on hyperbolic surfaces." pith.science (2026). https://pith.science/paper/NKKJTKL5

@misc{pith2026250612465,
  author       = {Pith},
  title        = {Pith review of: Shortest filling geodesics on hyperbolic surfaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NKKJTKL5}},
  note         = {Machine review of arXiv:2506.12465}
}
abstract

In this paper, we obtain the minimal length of a filling (multi-)geodesic on a genus $g$ hyperbolic surface in the moduli space of hyperbolic surfaces and show that it is realized by the geodesic whose complement is a right-angled regular $(8g-4)$-gon. A single geodesic realizing this minimum is provided.

Figures

Figures reproduced from arXiv: 2506.12465 by the authors.

Figure 1
Figure 1. Vertices of convex graphs (1) every trivalent vertex contains two opposite edges; (2) every 4-valent vertex is the intersection of two geodesic segments. A convex graph G is filling if every connected component of X \G is a polygon. If G ⊂ X is convex, then X \ G is a convex surface [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. A boundary component of S bounding a disk outside S. boundary α of D is piecewise geodesic and has all inner angle Ai > π and by the Gauss-Bonnet theorem, Area(D) = Pn i=1(π − Ai) − 2π < 0, which is a contradiction. Hence α is essential in X. Let γ be the unique closed geodesic in X in the homotopy class of α. Suppose that α ∩ γ ̸= ∅. For the universal covering p : D → X, as illustrated in [PITH_FULL_IMAGE:figures/… view at source ↗
Figure 3
Figure 3. γe intersects αe lift γe of γ. Let Se be a lift of S with αe as a boundary component. Since S is convex, Se has inner angles less than π for vertices on αe. The deck transformation g corresponding to ⟨α⟩ ∈ π1(Σ) gives infinitely many intersection points of αe and γe. Some adjacent two intersection points gives a polygon D with 2 vertices V and W on γe, and n vertices on αe (other than V and W) with inner angles A1, … view at source ↗
Figures from the paper (10 more)
Figure 4
Figure 4. Figure 4: An annulus between α and γ. Note that α does not bound a disk. Proof of Theorem 2.1 in the simple case. Γ is a 4-valent graph on X. Take a simple closed geodesic c ∈ Γ and let G1 = c. G1 is clearly convex since X \ G1 has no vertices. In Γ \ G1, there are proper arcs w…
Figure 5
Figure 5. Figure 5: If s2 and G2 bound the disk D, then s2 and γ will bound the bigon D∗ since γ and G2 are disjoint, where D∗ is part of D. is embedded since s2 is simple, and s2 and γ cobound a bigon D∗ which is impossible. Hence s2 is essential in the sense that it does not co-bound a …
Figure 6
Figure 6. Figure 6: Cutting curves for an empty subgraph. cutting curve for (Θ, G) is a geodesic β : [a, b] → X in Θ such that β : (a, b) → X \ G is injective, β(a) ∈ G and β(b) ∈ G ∪ β((a, b)). There are exactly two types of cutting curves when G is nonempty as in [PITH_FULL_IMAGE:figur…
Figure 7
Figure 7. Figure 7: Cutting curves for a nonempty subgraph. image. A cutting curve for (Θ, G) is ∂-parallel if cobounds a disk with G, and essential otherwise. A ∂-parallel cutting curve of type VI can be found in [PITH_FULL_IMAGE:figures/full_fig_p006_7.png]
Figure 8
Figure 8. Figure 8: If s2 and G2 bound the immersed disk D (left), then s2 and γ will bound the immersed bigon D∗ (right). The deeply shadowed regions are used twice. γ cobound an (immersed) bigon D∗ since β ∩ γ ̸= ∅ and γ ∩ G2 = ∅. (This is more clear if viewed in the universal cover). I…
Figure 9
Figure 9. Figure 9: The intersection of β and γ in the universal cover. We show that s2 is essential. If β ′ is simple, then s2 = β ′ is of type V and essential. If β ′ is not simple, then s2 is of type VI. If s2 is not essential, then s2 are ∂-parallel as in [PITH_FULL_IMAGE:figures/ful…
Figure 10
Figure 10. Figure 10: A ∂-parallel cutting curve of type VI. The green part of s2 will be referred to as the “circle” part. of the essential geodesic proper arc β ′ for X \ G2. As in [PITH_FULL_IMAGE:figures/full_fig_p008_10.png]
Figure 11
Figure 11. Figure 11: A ∂-parallel cutting curve in D. se2 of s2 is the starting part of a lift βe′ of β ′ . The rest part of βe′ will not intersect with any lift of the “circle” part of s2 since there will be a bigon between different lifts of β ′ . Thus βe′ and Gf2 will cobound a disk an…
Figure 12
Figure 12. Figure 12: A closed filling geodesic with (2g − 1) self-intersections. For g > 2, the part in the dashed rectangle is the building block and there are g − 3 copies. The double points separate α into edges labelled along the curve as in (B). For conciseness, we only label some ed…
Figure 13
Figure 13. Figure 13: Gluing pattern for genus 2 [PITH_FULL_IMAGE:figures/full_fig_p016_13.png]

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