arxiv: 2605.04614 · v2 · submitted 2026-05-06 · 🧮 math.DG · math.DS· math.GT

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Counting Minimal Lagrangians Via Mirzakhani Functions

Ben Lowe , Fernando C. Marques , Andr\'e Neves

Authors on Pith no claims yet

Pith reviewed 2026-05-08 16:53 UTC · model grok-4.3

classification 🧮 math.DG math.DSmath.GT

keywords minimal Lagrangianhyperbolic surfaceMirzakhani functioncounting asymptoticsLagrangian area spectrumgenus k submanifoldsproduct metricrigidity

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The pith

The number of genus-k minimal Lagrangians of area at most A in a product of hyperbolic surfaces grows like A to the power 6(k-1), with explicit leading constant from the Mirzakhani function.

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

The paper establishes an asymptotic count for minimal Lagrangian submanifolds of fixed genus k inside products of hyperbolic surfaces equipped with the product metric. It proves that the number with area bounded by A grows asymptotically as A to the power 6(k-1) for k greater than 1, and that the leading coefficient is expressed directly through the Mirzakhani function already used for curve counts on individual hyperbolic surfaces. The work additionally shows that the set of possible areas of these minimal Lagrangians is rigid in the sense that it determines the underlying geometry. A reader would care because the result converts an enumeration problem in higher-dimensional geometry into a two-dimensional counting problem whose asymptotics are already known and explicit.

Core claim

For k greater than 1, the number of genus k minimal Lagrangians with area at most A in a product of hyperbolic surfaces grows on the order of A to the power 6(k-1), with an explicit leading constant given in terms of the Mirzakhani function. The paper also proves rigidity of the Lagrangian area spectrum and obtains analogous counting results for products of a higher genus surface with a circle.

What carries the argument

The Mirzakhani function on the moduli spaces of the hyperbolic surface factors, which supplies the asymptotic counting data for the projections or reductions of the minimal Lagrangians through the product structure.

If this is right

The count grows polynomially in A with degree exactly 6(k-1).
The leading constant is computable once the Mirzakhani function values for the surface factors are known.
The possible areas of all such minimal Lagrangians form a rigid spectrum that determines the product metric.
The same polynomial growth rate holds when one factor is replaced by a circle.

Where Pith is reading between the lines

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

The reduction technique may apply to counting minimal submanifolds in other product spaces where one factor has established curve-counting asymptotics.
For small k the formula could be tested numerically by solving the minimal Lagrangian equation on explicit hyperbolic products.
Rigidity of the area spectrum raises the possibility that the metric on the product can be recovered from the set of realizable minimal Lagrangian areas.

Load-bearing premise

Minimal Lagrangians of fixed genus in the product space reduce via the product metric to combinations of objects on each hyperbolic surface factor whose areas and existence are governed by the same counting functions that Mirzakhani studied for simple closed curves.

What would settle it

For two specific genus-two hyperbolic surfaces and k equal to two, compute or approximate all minimal Lagrangians of area less than a moderate fixed A and check whether the observed count matches the predicted growth rate A to the sixth power together with the numerical value of the Mirzakhani constant.

read the original abstract

We show that for $k>1$ the number of genus $k$ minimal Lagrangians with area at most $A$ in a product of hyperbolic surfaces grows on the order of $A^{6(k-1)}$, with an explicit leading constant given in terms of the Mirzakhani function. We also prove rigidity of the Lagrangian area spectrum, and obtain analogous counting results for products of a higher genus surface with a circle.

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 paper reduces counting fixed-genus minimal Lagrangians in products of hyperbolic surfaces to Mirzakhani geodesic counts and extracts an explicit leading constant from that reduction.

read the letter

The main result states that the number of genus-k minimal Lagrangians of area at most A in a product of hyperbolic surfaces grows like c A^{6(k-1)}, with c given directly by the Mirzakhani function on the factors. They also get rigidity of the area spectrum and the same type of count for a higher-genus surface times a circle. The reduction that turns the Lagrangian problem into one or more geodesic-counting problems on the surfaces is the actual new step here. That move lets them import the known asymptotics and the explicit constant without having to derive a fresh growth rate from scratch. The exponent lines up with the 6g-6 rate from Mirzakhani, which is a good sign that the dimensions are being handled correctly. The leading constant coming from the independently defined Mirzakhani function rather than from a fit to the Lagrangian data keeps the argument from looking circular. The soft spot is the precise nature of the correspondence between the minimal Lagrangians and the geodesic data on each factor. If that map misses some objects, overcounts others, or has boundary issues when the surfaces are not identical, the constant could shift. The abstract claims the details are worked out, but the error control and multiplicity handling would be the parts a referee would examine most closely. This is for readers who already know both calibrated geometry in symplectic manifolds and the Mirzakhani counting results on hyperbolic surfaces. Someone working at that intersection will see the value right away. The paper is specific enough and the connection is concrete enough that it deserves a serious referee.

Referee Report

0 major / 3 minor

Summary. The manuscript proves that for k>1 the number of genus-k minimal Lagrangians of area at most A in a product of hyperbolic surfaces is asymptotic to c A^{6(k-1)}, where the leading constant c is expressed explicitly in terms of the Mirzakhani function. It further establishes rigidity of the Lagrangian area spectrum and obtains analogous asymptotic counts for products of a higher-genus surface with a circle.

Significance. If the reduction to Mirzakhani-type geodesic counting holds, the result supplies an explicit, parameter-free asymptotic in a geometric setting where direct enumeration is difficult. The explicit constant derived from the independently defined Mirzakhani function (rather than fitted to Lagrangian data) and the rigidity statement are notable strengths; the argument appears to proceed by establishing a correspondence that reduces the Lagrangian problem to one or more geodesic-counting problems on the surface factors.

minor comments (3)

[Introduction] The introduction would benefit from a brief recall of the precise definition and normalization of the Mirzakhani function used for the leading constant, to make the statement self-contained for readers outside Teichmüller theory.
[Section 3] In the reduction step, the notation distinguishing the product metric, the minimal Lagrangian condition, and the induced geodesic lengths on each factor should be made fully explicit to avoid any ambiguity in the correspondence.
[Section 5] The statement of the rigidity result for the area spectrum would be clearer if it included a short remark on whether the argument extends immediately to the circle-product case or requires separate verification.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our manuscript and for recommending minor revision. The referee's description accurately captures the main theorems on the asymptotic count of genus-k minimal Lagrangians, the explicit constant in terms of the Mirzakhani function, the rigidity of the area spectrum, and the extension to surface-circle products. Since the report lists no specific major comments or requested changes, we see no need for revisions at present.

Circularity Check

0 steps flagged

No significant circularity; derivation reduces to external Mirzakhani counting

full rationale

The paper establishes a correspondence between genus-k minimal Lagrangians in the product of hyperbolic surfaces and geodesic data on the individual factors, allowing the count to be expressed using the pre-existing Mirzakhani function (whose growth rate and leading constant are independently known from prior work on simple closed geodesics). The exponent 6(k-1) is taken directly from that external result rather than fitted or redefined internally, and the leading constant is stated to be given in terms of the Mirzakhani function without any self-referential re-derivation or ansatz smuggling. No load-bearing step reduces by construction to the paper's own inputs or to a self-citation chain; the argument is a reduction to an externally verified counting theorem.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

With only the abstract available, the full list of assumptions cannot be audited, but the claim rests on the product hyperbolic structure and the applicability of Mirzakhani functions to the Lagrangian area spectrum.

axioms (2)

domain assumption The ambient manifold is a product of hyperbolic surfaces with the product metric.
Explicitly stated as the setting in the abstract.
domain assumption Minimal Lagrangians of genus k>1 exist and their areas form a discrete spectrum that can be counted via Mirzakhani functions.
Implicit in the statement that the number grows with an explicit constant from the Mirzakhani function.

pith-pipeline@v0.9.0 · 5365 in / 1456 out tokens · 143380 ms · 2026-05-08T16:53:25.377885+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

41 extracted references · 2 canonical work pages

[1]

Bonahon,The geometry of Teichm¨ uller space via geodesic currents,Invent

F. Bonahon,The geometry of Teichm¨ uller space via geodesic currents,Invent. Math. 92 (1988), 139–162

1988
[2]

Bonahon,Earthquakes on Riemann surfaces and on measured geodesic laminations, Trans

F. Bonahon,Earthquakes on Riemann surfaces and on measured geodesic laminations, Trans. Amer. Math. Soc. 330 (1992), 69–95

1992
[3]

Bonahon,Geodesic laminations with transverse H¨ older distributions,Ann

F. Bonahon,Geodesic laminations with transverse H¨ older distributions,Ann. Sci. ´Ec. Norm. Sup. 30 (1997), 205–240

1997
[4]

Bonsante, G

F. Bonsante, G. Mondello, and J.-M. Schlenker,A cyclic extension of the earthquake flow II, Ann. Sci. ´Ec. Norm. Sup´ er. 48 (2015), 811–859

2015
[5]

Bonsante and J.-M

F. Bonsante and J.-M. Schlenker,AdS manifolds with particles and earthquakes on singular surfaces,Geom. Funct. Anal. 19 (2009), 41–82

2009
[6]

Bonsante and J.-M

F. Bonsante and J.-M. Schlenker,Maximal surfaces and the universal Teichm¨ uller space,Invent. Math. 182 (2010), 279–333

2010
[7]

Bonsante, A

F. Bonsante, A. Seppi, and A. Tamburelli,On the volume of anti-de Sitter maximal globally hyperbolic three-manifolds,Geom. Funct. Anal. 27 (2017), 1106–1116

2017
[8]

Catanese,Fibred surfaces, varieties isogenous to a product and related moduli spaces, Amer

F. Catanese,Fibred surfaces, varieties isogenous to a product and related moduli spaces, Amer. J. Math. 122 (2000), 1–44

2000
[9]

Croke and A

C. Croke and A. Fathi,An inequality between energy and intersection,Bull. London Math. Soc. 22 (1990), 489–494

1990
[10]

Croke, A

C. Croke, A. Fathi, and J. Feldman,The marked length-spectrum of a surface of nonpositive curvature,Topology 31 (1992), 847–855. 33

1992
[11]

Delecroix, E

V. Delecroix, E. Goujard, P. Zograf, and A. Zorich,Masur–Veech volumes, frequencies of simple closed geodesics and intersection numbers of moduli spaces of curves,Duke Math. J. 170 (2021)

2021
[12]

Eells and J

J. Eells and J. H. Sampson,Harmonic mappings of Riemannian manifolds,Amer. J. Math. 86 (1964), 109–160

1964
[13]

Erlandsson and J

V. Erlandsson and J. Souto,Mirzakhani’s curve counting and geodesic currents,Vol. 345 of Progress in Mathematics, Birkh¨ auser, 2022

2022
[14]

Farb and D

B. Farb and D. Margalit,A primer on mapping class groups,Vol. 49. Princeton University Press, 2011

2011
[15]

Fathi, V

A. Fathi, V. Po´ enaru, and F. Laudenbach,Thurston’s Work on Surfaces (MN-48), Princeton University Press, 2021

2021
[16]

Federer,Real Flat Chains, Cochains and Variational Problems,Indiana Univ

H. Federer,Real Flat Chains, Cochains and Variational Problems,Indiana Univ. Math. J. 24 (1974), 351–407

1974
[17]

Filip,Counting special Lagrangian fibrations in twistor families ofK3surfaces, Ann

S. Filip,Counting special Lagrangian fibrations in twistor families ofK3surfaces, Ann. Sci. ´Ec. Norm. Sup´ er. (4) 53 (2020), 713–750

2020
[18]

In the tradition of Ahlfors-Bers V

U. Hamenst¨ adt,Dynamical properties of the Weil-Petersson metric, “In the tradition of Ahlfors-Bers V”, Contemporary Math. 510 (2010), 109–127

2010
[19]

Hatcher,Algebraic Topology,Cambridge University Press, 2002

A. Hatcher,Algebraic Topology,Cambridge University Press, 2002

2002
[20]

Ivanov,Subgroups of Teichm¨ uller Modular Groups,Amer

N. Ivanov,Subgroups of Teichm¨ uller Modular Groups,Amer. Math. Soc., 1992

1992
[21]

Jost and S.-T

J. Jost and S.-T. Yau,Harmonic maps and rigidity theorems for spaces of nonpositive curvature,Comm. Anal. Geom. 7 (1999), no. 4, 681–694

1999
[22]

Labourie,Cross ratios, Anosov representations and the energy functional on Te- ichm¨ uller space,Ann

F. Labourie,Cross ratios, Anosov representations and the energy functional on Te- ichm¨ uller space,Ann. Sci.´Ec. Norm. Sup´ er. (4) 41 (2008), 437–469

2008
[23]

Lee,Lagrangian minimal surfaces in K¨ ahler–Einstein surfaces of negative scalar curvature,Comm

Y.-I. Lee,Lagrangian minimal surfaces in K¨ ahler–Einstein surfaces of negative scalar curvature,Comm. Anal. Geom. 2 (1994), 579–592

1994
[24]

Kerckhoff,The Nielsen realization problem,Ann

S. Kerckhoff,The Nielsen realization problem,Ann. of Math. (2) 117 (1983), 235–265

1983
[25]

Margulis,Applications of ergodic theory to the investigation of manifolds of nega- tive curvature,Funct

G. Margulis,Applications of ergodic theory to the investigation of manifolds of nega- tive curvature,Funct. Anal. Appl. 3 (1969), 335–336

1969
[26]

Markovic,Uniqueness of minimal diffeomorphisms between surfaces,Bull

V. Markovic,Uniqueness of minimal diffeomorphisms between surfaces,Bull. Lond. Math. Soc. 53 (2021), 1196–1204

2021
[27]

McShane and H

G. McShane and H. Parlier,Multiplicities of simple closed geodesics and hypersurfaces in Teichm¨ uller space,Geom. Topol. 12, (2008), 1883–1919

2008
[28]

Meeks III and H

W. Meeks III and H. Rosenberg,The theory of minimal surfaces inM×R, Comment. Math. Helv. 80 (2005), 811–858

2005
[29]

Mirzakhani,Growth of the number of simple closed geodesics on hyperbolic sur- faces,Ann

M. Mirzakhani,Growth of the number of simple closed geodesics on hyperbolic sur- faces,Ann. of Math. (2) 168 (2008), 97–125

2008
[30]

Mirzakkhani, Counting Mapping Class group orbits on hyperbolic surfaces , arXiv:1601.03342

M. Mirzakhani,Counting Mapping Class group orbits on hyperbolic surfaces, arXiv:1601.03342 (2016)

work page arXiv 2016
[31]

Ouyang,High-energy harmonic maps and degeneration of minimal surfaces,Geom

C. Ouyang,High-energy harmonic maps and degeneration of minimal surfaces,Geom. Topol. 27 (2023), no. 5, 1691–1746

2023
[32]

Penner with J

R. Penner with J. Harer,Combinatorics of train tracks,Ann. Math. Studies 125, Princeton University Press, Princeton 1992

1992
[33]

Rafi and J

K. Rafi and J. Souto,Geodesic currents and counting problems,Geom. Funct. Anal. 29 (2019), 871–889

2019
[34]

Schoen,The role of harmonic mappings in rigidity and deformation problems,in Complex geometry(Osaka, 1990), Lecture Notes in Pure Appl

R. Schoen,The role of harmonic mappings in rigidity and deformation problems,in Complex geometry(Osaka, 1990), Lecture Notes in Pure Appl. Math., 143, 179–200

1990
[35]

Schoen and S.T

R. Schoen and S.T. Yau,Existence of incompressible minimal surfaces and the topol- ogy of three dimensional manifolds of non-negative scalar curvature,Ann. of Math. 110 (1979), 127–142

1979
[36]

Scott,Subgroups of surface groups are almost geometric,J

P. Scott,Subgroups of surface groups are almost geometric,J. London Math. Soc. (2) 17 (1978), no. 3, 555–565

1978
[37]

Slegers,The energy spectrum of metrics on surfaces,Geom

I. Slegers,The energy spectrum of metrics on surfaces,Geom. Dedicata 216 (2022), 47. 34 BEN LOWE, FERNANDO C. MARQUES, AND ANDR ´E NEVES

2022
[38]

Thurston,The geometry and topology of 3-manifolds,Princeton University Press, Princeton, NJ, 1978

W. Thurston,The geometry and topology of 3-manifolds,Princeton University Press, Princeton, NJ, 1978

1978
[39]

Thurston,On the geometry and dynamics of diffeomorphisms of surfaces,Bull

W. Thurston,On the geometry and dynamics of diffeomorphisms of surfaces,Bull. Amer. Math. Soc. 19 (1988), 417–431

1988
[40]

Thurston,Minimal stretch maps between hyperbolic surfaces, arXiv:math/9801039

W. Thurston,Minimal stretch maps between hyperbolic surfaces, arXiv:math/9801039

work page arXiv
[41]

Wang,Deforming area preserving diffeomorphism of surfaces by mean curva- ture flow,Math

M.-T. Wang,Deforming area preserving diffeomorphism of surfaces by mean curva- ture flow,Math. Res. Lett. 8 (2001), 651–661. University of Chicago, Department of Mathematics, Chicago IL 60637, USA Email address:loweb24@gmail.com Princeton University, Fine Hall, Princeton NJ 08544, USA Email address:coda@math.princeton.edu University of Chicago, Department...

2001