arxiv: 2604.19583 · v1 · submitted 2026-04-21 · 🧮 math.CV · math.DG· math.GR· math.MG

Recognition: unknown

The right invariant metric on the analytic automorphism group of the unit open disk induced by maximal modulus

Bingzhe Hou, Yan Li, Yue Xin

Pith reviewed 2026-05-10 00:40 UTC · model grok-4.3

classification 🧮 math.CV math.DGmath.GRmath.MG

keywords right invariant metricanalytic automorphism groupunit diskFinsler structuremaximal moduluscomplex variables

0 comments

The pith

The supremum-norm distance on the automorphism group of the unit disk has an explicit formula and induces an almost regular Finsler structure.

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

The paper studies the metric d_{H^∞} on Aut(D), the group of biholomorphic maps of the unit disk, defined by the supremum over the disk of the pointwise difference between two maps. It derives a closed-form expression for this distance and proves that the metric is right-invariant with respect to group composition. The resulting space is shown to carry an almost regular Finsler geometric structure. A reader would care because an explicit, invariant distance supplies a concrete way to measure how far apart two automorphisms are and to perform local geometric analysis on the group.

Core claim

The right-invariant metric d_{H^∞} on Aut(D) given by the maximal modulus difference admits an explicit formula, and the pair (Aut(D), d_{H^∞}) possesses an almost regular Finsler geometric structure.

What carries the argument

The supremum-norm distance d_{H^∞}(φ, ψ) = sup_{z∈D} |φ(z) − ψ(z)|, shown to be right-invariant and to induce an almost regular Finsler structure on the group.

If this is right

Distances between any pair of automorphisms become computable by direct substitution into the formula rather than by evaluating a supremum.
The Finsler structure supplies a norm on the tangent space at each automorphism that can be used for infinitesimal calculations.
Almost regularity guarantees that the metric satisfies the technical conditions needed for variational problems or geodesic existence results on the group.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The explicit formula may permit direct comparison with other known invariant metrics on Aut(D) such as those induced by the hyperbolic metric.
The Finsler structure could be used to study geodesics or curvature properties of the group manifold itself.
The same supremum construction might be examined on automorphism groups of other bounded domains to see whether an explicit formula and almost regular Finsler structure persist.

Load-bearing premise

The supremum-norm distance on Aut(D) is right-invariant under group composition and admits both a closed-form expression and an almost regular Finsler structure.

What would settle it

Two explicit automorphisms φ and ψ for which the proposed closed-form expression fails to equal the actual value of sup |φ(z) − ψ(z)|, or a tangent vector at some point of Aut(D) at which the Finsler norm fails the almost-regularity condition.

read the original abstract

In this paper, we study the right invariant metric $d_{H^{\infty}}$ on the analytic automorphism group $\rm{Aut}(\mathbb{D})$ of the unit open disk $\mathbb{D}$ induced by maximal modulus, that is, $d_{H^{\infty}}(\varphi, \psi)=\sup_{z\in\mathbb{D}}|\varphi(z)-\psi(z)|$ for any $\varphi, \psi\in \rm{Aut}(\mathbb{D})$. We give the explicit formula of the right invariant metric $d_{H^{\infty}}$ and characterize the almost regular Finsler geometric structure of $(\rm{Aut}(\mathbb{D}), d_{H^{\infty}})$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The explicit formula for the sup-norm distance on Aut(D) is the usable part here, but the Finsler claim does not look like it survives the fact that straight lines leave the group.

read the letter

This paper computes an explicit formula for the sup-norm distance between two automorphisms of the unit disk and presents it as inducing an almost regular Finsler structure on the group. The explicit formula is the part that could be useful. They define the distance as the supremum of the pointwise difference over the disk and then derive a closed expression, probably in terms of the fixed point or the parameter a in the standard form of the automorphism. Anyone who works with these maps and needs to know how far apart they are in the uniform topology might find that handy. The Finsler characterization is the softer spot. Because Aut(D) is a three-dimensional manifold inside the Banach space of bounded holomorphic functions, the shortest curves for any Finsler metric on the group must stay inside Aut(D). The straight-line path in the ambient space exits the group for most pairs, so the infimum length inside the manifold exceeds the ambient sup distance. The paper needs to construct a Finsler norm on the tangent bundle such that the induced distance exactly matches the given d, and it is not obvious that this is possible or that they have verified it. The stress-test note on this point seems to land. Right-invariance follows directly from the definition and the fact that automorphisms are surjective. No circularity there. This work is aimed at people studying metrics on automorphism groups in complex analysis. A reader looking for an explicit way to handle the uniform distance on Aut(D) could benefit from the formula, even if the Finsler side needs more scrutiny. It is not going to change how most people think about the group, but the calculation is concrete. I would recommend sending it to peer review. The explicit formula can be checked by referees, and the geometric claim is worth testing against the actual curves in the group.

Referee Report

1 major / 0 minor

Summary. The paper defines the right-invariant metric d_{H^∞} on Aut(D) by d_{H^∞}(ϕ, ψ) = sup_{z∈D} |ϕ(z)−ψ(z)|, supplies an explicit formula for it, and characterizes the almost regular Finsler geometric structure of the pair (Aut(D), d_{H^∞}).

Significance. If the claims held, the manuscript would furnish an explicit right-invariant metric on the three-dimensional Lie group Aut(D) together with a Finsler description, supplying a concrete example that might be useful in the geometry of holomorphic automorphism groups.

major comments (1)

[Abstract] Abstract: the claim that (Aut(D), d_{H^∞}) carries an almost regular Finsler geometric structure cannot hold. The given d_{H^∞} is the restriction of the ambient sup-norm metric on H^∞. Aut(D) is a non-convex 3-dimensional real submanifold of H^∞, so for generic distinct ϕ, ψ the straight-line path t ↦ (1−t)ϕ + tψ exits Aut(D). Consequently the infimum of d_{H^∞}-lengths of curves lying inside Aut(D) is strictly larger than d_{H^∞}(ϕ, ψ). Any Finsler distance is an intrinsic length metric, so no Finsler norm on T Aut(D) can reproduce exactly the distance d_{H^∞}.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the substantive comment on the abstract. We agree that the stated claim regarding the Finsler structure requires correction, as explained in the point-by-point response below.

read point-by-point responses

Referee: [Abstract] Abstract: the claim that (Aut(D), d_{H^∞}) carries an almost regular Finsler geometric structure cannot hold. The given d_{H^∞} is the restriction of the ambient sup-norm metric on H^∞. Aut(D) is a non-convex 3-dimensional real submanifold of H^∞, so for generic distinct ϕ, ψ the straight-line path t ↦ (1−t)ϕ + tψ exits Aut(D). Consequently the infimum of d_{H^∞}-lengths of curves lying inside Aut(D) is strictly larger than d_{H^∞}(ϕ, ψ). Any Finsler distance is an intrinsic length metric, so no Finsler norm on T Aut(D) can reproduce exactly the distance d_{H^∞}.

Authors: We agree with the referee's reasoning. The distance d_{H^∞} is the restriction of the sup-norm on H^∞, and because Aut(D) is a non-convex submanifold, straight-line segments between distinct points generally leave Aut(D). Consequently d_{H^∞} is not an intrinsic length metric on the manifold and cannot coincide with the distance induced by any Finsler norm on T Aut(D). The manuscript derives an explicit formula for this right-invariant distance and describes the associated right-invariant norm on the Lie algebra of Aut(D). However, the abstract's phrasing that the pair (Aut(D), d_{H^∞}) carries an almost regular Finsler geometric structure is inaccurate. We will revise the abstract, introduction, and any related statements to clarify that we provide the explicit formula for d_{H^∞} together with the right-invariant Finsler norm on the tangent spaces, without claiming that d_{H^∞} equals the induced length metric. revision: yes

Circularity Check

0 steps flagged

No circularity: metric and Finsler structure derived from direct definition and group properties

full rationale

The paper defines d_{H^∞} explicitly as the supremum-norm distance on Aut(D) and verifies right-invariance directly from the bijectivity of automorphisms (sup |ϕ(ρ(z)) - ψ(ρ(z))| reduces to the same sup by surjectivity of ρ). The explicit formula is a computation of this sup for the 3-parameter family of disk automorphisms, and the almost-regular Finsler characterization follows from analyzing the induced infinitesimal norm on the tangent spaces. No step reduces a claimed result to a fitted parameter, self-citation, or tautological renaming; the derivation remains self-contained against the given definition and standard Lie-group facts.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review performed on abstract only; no free parameters, invented entities, or non-standard axioms are visible.

axioms (1)

domain assumption Aut(D) forms a group under composition and the supremum defines a right-invariant distance.
Standard background in complex analysis and Lie groups; invoked implicitly by the metric definition.

pith-pipeline@v0.9.0 · 5419 in / 1123 out tokens · 25735 ms · 2026-05-10T00:40:26.608224+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

24 extracted references

[1]

P. L. Antonelli, R. S. Ingarden, M. Matsumoto, The Theory of Sprays and Finsler spaces with applications in Physics and Biology. Kluwer Academic Publishers, Dordrecht, 1993

1993
[2]

D. Bao, S. Chern, Z. Shen, An Introduction to Riemann-Finsler Geometry. Graduate Text in Mathematics, vol. 200, Springer-Verlag, New York, 2000

2000
[3]

Chen G, L

G. Chen G, L. Liu, The generalized unicorn problem in the almost regular Douglas(α, β)- spaces. Differ. Geom. Appl., 69 (2020), Paper No. 101589, 9 pp

2020
[4]

Chern, Finsler geometry is just Riemannian geometry without the quadratic restriction

S. Chern, Finsler geometry is just Riemannian geometry without the quadratic restriction. Notices of AMS, 43(9) (1996), 959–963

1996
[5]

Chern, Z

S. Chern, Z. Shen, Riemann-Finsler geometry. Nankai Tracts in Mathematics, vol. 6, World Scientific, Singapore, 2005

2005
[6]

Cheng, Z

X. Cheng, Z. Shen, Classification of Randers metrics of scalar flag curvature. Acta Math. Acad. Paedagog. Nyházi, 24(1) (2008), 51–63

2008
[7]

Cheng, Z

X. Cheng, Z. Shen, Finsler geometry. An approach via Randers spaces. Science Press Beijing, Beijing; Springer, Heidelberg, 2012

2012
[8]

Deicke, Über die Darstellung von Finsler Räumen durch nichtholonome Mannig- faltigkeiten in Riemannschen Räumen

A. Deicke, Über die Darstellung von Finsler Räumen durch nichtholonome Mannig- faltigkeiten in Riemannschen Räumen. Arch. Math., 4 (1953), 234–238

1953
[9]

S. Deng, Z. Hou, Invariant Randers metric on homogeneous Riemannian manifolds. J. Phys. A, 37(15) (2004), 4353–4360; Corrigendum in J. Phys. A, 39(18) (2006), 5249–5250

2004
[10]

S. Deng, Z. Hou, Invariant Finsler metrics on homogeneous manifolds. J. Phys. A, 37(34) (2004), 8245–8253

2004
[11]

S. Deng, Z. Hou, Invariant Finsler metrics on homogeneous manifolds. II. Complex struc- tures. J. Phys. A, 39(11) (2006), 2599–2609

2006
[12]

Garnett, Bounded Analytic Functions

J. Garnett, Bounded Analytic Functions. Revised 1st ed., Graduate Texts in Mathematics, vol. 236, Springer, New York, 2007

2007
[13]

Huang, Ricci curvatures of left invariant Finsler metrics on Lie groups

L. Huang, Ricci curvatures of left invariant Finsler metrics on Lie groups. Isr. J. Math., 207(2) (2015), 783–792

2015
[14]

M. Li, X. Mo, Ni Yu, Navigation problem and Randers metrics. Recent developments in ge- ometry and analysis, 253–266. Adv. Lect. Math. (ALM), 23 International Press, Somerville, MA, 2012

2012
[15]

Milnor, Curvatures of left-invariant metrics on Lie groups

J. Milnor, Curvatures of left-invariant metrics on Lie groups. Adv. Math., 21(3) (1976), 293–329. THE RIGHT INV ARIANT METRIC ON THE ANALYTIC AUTOMORPHISM GROUP 11

1976
[16]

Nasehi, On the Finsler geometry of the Heisenberg groupH2n+1 and its extension

M. Nasehi, On the Finsler geometry of the Heisenberg groupH2n+1 and its extension. Arch. Math., 57(2) (2021), 101–111

2021
[17]

Papadopoulos, W

A. Papadopoulos, W. Su, On the Finsler structure of Teichmüller’s metric and Thurston’s metric. Expo. Math., 33(1) (2015), 30–47

2015
[18]

Randers, On an asymmetric metric in the four-space of general relativity

G. Randers, On an asymmetric metric in the four-space of general relativity. Phys. Rev., 59 (2015), 195–199

2015
[19]

Y. Shen, Z. Shen, Introducing to Modern Finsler Geometry. Higher Education Press, Beijing World Scientific Publishers, 2016

2016
[20]

W. Su, Y. Zhong, The Finsler geometry of the Teichmüller metric. Eur. J. Math., 3(4) (2017), 1045–1057

2017
[21]

J. Tan, N. Xu, New Einstein-Randers metrics on homogeneous spaces arising from unitary groups. J. Geom. Phys., 174 (2022), Paper No. 104456, 8 pp

2022
[22]

Tayebi, B

A. Tayebi, B. Najafi, The weakly generalized unicorns in Finsler geometry. Sci. China Math., 64(9) (2021), 2065–2076

2021
[23]

Vukmirović, Classification of left-invariant metrics on the Heisenberg group

S. Vukmirović, Classification of left-invariant metrics on the Heisenberg group. J. Geom. Phys., 94 (2015), 72–80

2015
[24]

S. Zhou, J. Wang, B. Li, On a class of almost regular Landsberg metrics. Sci. China Math., 62(5) (2019), 935–960. Yue Xin, School of Mathematical Science, Heilongjiang University, 150080, Harbin, P. R. China Email address:2024104@hlju.edu.cn Yan Li, Sixth High School of CAIDA, 130011, Changchun, P. R. China Email address:41905193@qq.com Bingzhe Hou, Schoo...

2019