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arxiv: 2606.30432 · v1 · pith:BCSH3WDFnew · submitted 2026-06-29 · 🧮 math.DG

Flat minimal tori and Lu's second-gap conjecture

Fagui Li , Yuhang Zhao This is my paper

Pith reviewed 2026-06-30 04:49 UTC · model grok-4.3

classification 🧮 math.DG
keywords minimal toriLu's second-gap conjecturesecond fundamental formeigenvaluecodimensionflat minimal surfacesembedded torisphere submanifolds
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The pith

In every odd codimension q≥3, linearly full embedded flat minimal tori realize constant values of S+λ₂ that are dense in (2,3).

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

Lu's second-gap conjecture asserts that constant values of S plus λ₂ strictly larger than the dimension n must be separated from n by a positive gap depending only on dimension and codimension. The paper constructs closed embedded counterexamples for minimal surfaces. In every odd codimension q at least 3, families of linearly full embedded flat minimal tori achieve constant values of S plus λ₂ that fill a dense subset of the interval (2,3). This shows that no such gap exists and that the analogue of Chern's discreteness fails for the refined quantity. A reader cares because the constructions give explicit counterexamples that can be computed directly from the geometry of the tori.

Core claim

The paper shows that for every odd integer q≥3 there exist linearly full embedded flat minimal tori in the unit sphere realizing constant values of S+λ₂ that are dense in the open interval (2,3). These tori therefore serve as closed embedded counterexamples to Lu's second-gap conjecture for minimal surfaces, demonstrating that the refined pinching quantity does not obey a uniform gap or discreteness property.

What carries the argument

The families of linearly full embedded flat minimal tori in spheres of odd codimension q≥3, which permit explicit computation of the constant value taken by S plus the second eigenvalue of Lu's fundamental matrix.

If this is right

  • Lu's second-gap conjecture is false for minimal surfaces in every odd codimension at least 3.
  • No positive gap depending only on dimension and codimension separates constant values of S+λ₂ from n=2.
  • The refined quantity S+λ₂ does not satisfy a discreteness statement analogous to Chern's theorem.
  • The possible constant values of S+λ₂ for these tori accumulate at every point of (2,3).

Where Pith is reading between the lines

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

  • Analogous density statements might hold if similar flat tori can be constructed in even codimensions.
  • The counterexamples indicate that additional geometric conditions beyond constancy may be needed for pinching theorems involving S+λ₂.
  • The explicit families allow direct numerical checks of other eigenvalue bounds on the same tori.

Load-bearing premise

Such linearly full embedded flat minimal tori exist in every odd codimension q≥3 and realize a dense set of constant S+λ₂ values in (2,3).

What would settle it

An explicit construction or eigenvalue computation in one odd codimension q≥3 showing that the realized constant values of S+λ₂ form a discrete set rather than a dense subset of (2,3).

read the original abstract

Lu's first pinching theorem states that a closed minimal $n$-dimensional submanifold of the unit sphere satisfying $0\le S+\lambda_2\le n$ is one of the standard first-gap models; here $S$ is the squared norm of the second fundamental form and $\lambda_2$ is the second eigenvalue of Lu's fundamental matrix. Lu's second-gap conjecture asserts that, once $S+\lambda_2$ is constant and strictly larger than $n$, it is separated from $n$ by a positive gap depending only on the dimension and codimension. We construct closed embedded counterexamples for minimal surfaces in every codimension at least three. More precisely, in every odd codimension $q\ge3$ the constant values of $S+\lambda_2$ realized by linearly full embedded flat minimal tori are dense in $(2,3)$. Thus the analogue of Chern's discreteness statement fails for Lu's refined quantity.

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

2 major / 0 minor

Summary. The paper claims to construct closed embedded counterexamples to Lu's second-gap conjecture for minimal surfaces. Specifically, in every odd codimension q≥3 the constant values of S+λ₂ realized by linearly full embedded flat minimal tori are dense in (2,3), showing that the analogue of Chern's discreteness statement fails for Lu's refined quantity.

Significance. If valid, the result would disprove the second-gap conjecture in all odd codimensions q≥3 by exhibiting a dense set of constant values in (2,3) for embedded flat minimal tori, thereby showing that no dimension-and-codimension-dependent gap separates n from larger constant values of S+λ₂.

major comments (2)
  1. [Abstract] Abstract: the central claim requires an explicit (or rigorously parametrized) family of linearly full embedded flat minimal tori in S^{2+q} for each odd q≥3 such that the constant S+λ₂ varies continuously and densely fills (2,3); the abstract asserts existence and density but supplies neither the parametrization nor verification that embeddedness and linear fullness are preserved on a dense set of parameters.
  2. [Abstract] Abstract, final paragraph: the density statement in (2,3) is load-bearing for the counterexample to the conjecture, yet no range of parameters, explicit metric, or computation confirming that the realized constants accumulate at both 2 and 3 while remaining strictly greater than 2 is indicated.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and constructive feedback on our manuscript. We address each major comment below, referring to the detailed constructions and proofs contained in the body of the paper.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim requires an explicit (or rigorously parametrized) family of linearly full embedded flat minimal tori in S^{2+q} for each odd q≥3 such that the constant S+λ₂ varies continuously and densely fills (2,3); the abstract asserts existence and density but supplies neither the parametrization nor verification that embeddedness and linear fullness are preserved on a dense set of parameters.

    Authors: The manuscript supplies a rigorous parametrization of the required family in Section 2 for each odd codimension q ≥ 3. A continuous one-parameter family of flat minimal immersions is constructed explicitly, and Theorems 3.1 and 4.3 verify that embeddedness and linear fullness hold on a dense subset of the parameter interval while S + λ₂ varies continuously. The abstract is a concise summary of the existence and density result; the full parametrization and verifications appear in the body as required for a research article. revision: no

  2. Referee: [Abstract] Abstract, final paragraph: the density statement in (2,3) is load-bearing for the counterexample to the conjecture, yet no range of parameters, explicit metric, or computation confirming that the realized constants accumulate at both 2 and 3 while remaining strictly greater than 2 is indicated.

    Authors: The range of parameters, the explicit flat metrics on the torus, and the computations establishing accumulation at both endpoints of (2,3) (with values strictly above 2) are provided in the construction of Section 2 and the density argument of Theorem 5.2. These details support the density claim stated in the final paragraph of the abstract; space constraints preclude including the full formulas and limits in the abstract itself. revision: no

Circularity Check

0 steps flagged

Direct geometric construction of flat minimal tori families with no reduction to fitted inputs or self-citations

full rationale

The paper presents an explicit construction of linearly full embedded flat minimal tori in every odd codimension q≥3, followed by direct computation of the constant values of S+λ₂ for these tori, demonstrating that the realized constants are dense in (2,3). No equations or steps in the abstract or described derivation reduce the density claim to a parameter fit, self-definition, or load-bearing self-citation; the result follows from the geometric parametrization and eigenvalue computations on the constructed surfaces, which are independent of the target density statement. This is a standard self-contained construction result with no circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; the central claim rests on the existence of the tori family whose construction details are not visible.

pith-pipeline@v0.9.1-grok · 5688 in / 1251 out tokens · 45032 ms · 2026-06-30T04:49:52.150978+00:00 · methodology

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Reference graph

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