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

Counting Minimal Lagrangians Via Mirzakhani Functions

Ben Lowe , Fernando C. Marques , Andr\'e Neves This is my paper

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.

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)
  1. [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.
  2. [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.
  3. [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

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Reference graph

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