Lu's conjecture for minimal surfaces in codimension two

Fagui Li; Jianquan Ge; Yunheng Zhang

Lu's conjecture for minimal surfaces in codimension two

T0 review · 0 major / 5 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read In S⁴, a closed minimal surface with constant S+λ₂ must have value 0 or 2, and the paper classifies all such surfaces.

desk verdict This paper closes the last open codimension for Lu's second-gap conjecture on minimal surfaces with a clean and, as far as I can tell, correct classification. read the letter →

arxiv 2607.21336 v2 pith:TH24VJIF submitted 2026-07-23 math.DG

Jianquan Ge , Fagui Li , Yunheng Zhang This is my paper

classification math.DG MSC 53C2053C2453C42

keywords minimalsurfacesLu'sfundamentalmatrixsecond-gapconjecturerigiditytheoremS⁴CodazziequationsVeronesesurfaceCliffordtorus

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

The paper proves that for any closed minimal surface immersed in the unit 4-sphere, if the quantity S+λ₂ — the squared norm of the second fundamental form plus the smaller eigenvalue of Lu's fundamental matrix — is constant, then that constant is either 0 or 2. The constant 0 forces a totally geodesic 2-sphere; the constant 2 forces either a Clifford torus inside a totally geodesic 3-sphere or the Veronese surface. This excludes every constant value above 2, so Lu's second-gap conjecture holds for minimal surfaces in codimension two. Along with the known hypersurface case and the counterexamples in every higher codimension, this closes the classification question for surfaces.

What carries the argument

The central objects are Lu's fundamental matrix, whose eigenvalues λ₁≥λ₂ are the squared lengths of the shape operators, and the refined curvature quantity S+λ₂, which interpolates between hypersurface and higher-codimensional pinching. The key local mechanism is an explicit differential identity expressing the connection forms as exact multiples of ∗db on the set where λ₁>λ₂>0, obtained by solving the Codazzi equations with the constancy condition. This yields a strictly positive formula for |∇b|², preventing critical points of b. A second ingredient is the real analyticity of minimal immersions into spheres, which allows the identity theorem for analytic functions to rule out maxima on the

What would settle it

Construct a closed minimal surface in S⁴ with S+λ₂ constant equal to some value strictly between 2 and 3. If such a surface exists, the theorem is false; a direct computation of |∇b| on such a surface would already contradict the strictly positive formula (3.5) at a critical point of b in the simple-eigenvalue region.

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Extended reading notes

Core claim

On the open set where the two eigenvalues of Lu's fundamental matrix are simple, the constancy of S+λ₂ forces the connection forms ω12 and ω34 to be proportional to the differential of the local function b, with coefficients depending only on b. This local identity yields an explicit rational formula for |∇b|² which is strictly positive, proving that b has no critical point and hence λ₂ has no interior maximum on that set. A real-analytic continuation argument, valid because minimal immersions into spheres are real analytic, rules out a maximum on the double-eigenvalue locus. The remaining degenerate configurations λ₂≡0 and λ₁≡λ₂ are classified separately, and both force the constant to be 2

Load-bearing premise

The entire contradiction argument for c>2 depends on the discriminant (λ₁−λ₂)² being real-analytic on all of M, so that if it vanishes on a set with interior it vanishes everywhere; without this real-analytic continuation, a maximum of λ₂ on the double-eigenvalue locus could persist and the exclusion of c>2 would collapse.

Editorial extensions

If this is right

No closed minimal surface in S⁴ admits S+λ₂ constant and greater than 2; the only constant values are 0 and 2.
Lu's second-gap conjecture is true for minimal surfaces in codimension two, and together with earlier results the full codimension picture for surfaces is now known: the conjecture holds for codimensions 1 and 2 and fails in every codimension ≥3.
The rigidity is sharp: the Clifford torus and Veronese surface saturate the constant 2, while the totally geodesic sphere gives 0, and the classification leaves no other closed examples.
The proof's local differential identity may serve as a template for higher-dimensional gap theorems under eigenvalue-simplicity assumptions.
For nonorientable minimal surfaces in S⁴, the same classification holds after passing to the connected oriented double cover, so the rigidity does not depend on orientability.

Reading between the lines

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

The method suggests a testable route to Lu's conjecture in higher dimensions n≥3: a similar local identity might exclude interior maxima of λ₂ on the simple-eigenvalue set, with the main obstruction being control of the boundary where eigenvalues coalesce.
Constancy of S+λ₂ might be relaxable to a pinching hypothesis such as S+λ₂ ≥ n or ≤ n+ε; the same identity could yield a gap result without full rigidity when equality is absent.
The exclusion of c>2 relies essentially on real-analytic continuation; a purely smooth proof would need a new argument to cross the double-eigenvalue locus, and constructing a smooth—but not analytic—minimal surface with constant S+λ₂>2 would be a meaningful stress test of the conjecture's boundaries.
The local identity (3.5) predicts |∇b|>0 on the simple-eigenvalue set for every constant c>2, a fact that could be checked numerically on candidate surfaces even before attempting a global construction.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

I am fairly convinced by this paper. It proves the (2,2) case of Lu's second-gap conjecture for minimal surfaces: if a closed minimal surface in S^4 has constant S+λ2, the constant is 0 or 2, and the only examples are the totally geodesic sphere, the Clifford torus in a totally geodesic S^3, and the Veronese surface. No constant value above 2 occurs. That completes the picture for surfaces, since hypersurfaces were known and Li–Zhao have counterexamples in every codimension ≥3.

The genuinely new ingredient is the local differential identity in Lemma 3.2, an explicit formula for |∇b|^2 on the open set U where the two eigenvalues of Lu's fundamental matrix are distinct and positive. I checked the moving-frame algebra in Lemmas 3.1 and 3.2; the signs and factors are consistent, and the exclusion of the zero denominator c−8b^2 is legitimate. Once that identity is in hand, the maximum-principle argument is clean: at a maximum of λ2 that lies in U, the gradient must vanish, contradicting Lemma 3.2; if the maximum lies on the λ1=λ2 locus, a sequence of points in U approaching it forces λ2 ≤ c/4, contradicting c=3λ2 there. The degenerate cases λ2≡0 and λ1≡λ2 are handled by separate lemmas, and both force c=2.

The soft spots are minor. The proof is not machine-checked and relies on standard external results: Lu's first-gap theorem, the do Carmo–Wallach classification, and Morrey's analyticity theorem. The most delicate is Morrey, used in Lemma 4.1 to rule out an open set of umbilic points. The minimal surface PDE in isothermal coordinates is indeed an analytic strongly elliptic system, so the appeal is appropriate; the step where real-analytic orthonormal frames are chosen after the metric is known analytic is asserted rather than shown in detail, but it is true. A referee might ask for one clarifying sentence there.

The paper is honest about what it does: the title says codimension two, but the actual scope is minimal surfaces (n=2,m=2), and the introduction makes that explicit. The citation practice is normal; the self-citations are context, not load-bearing.

Bottom line: this deserves a serious referee and, assuming the analyticity step is accepted, publication. I would bring it to reading group.

Referee Report

0 major / 5 minor

Summary. The paper classifies closed minimal surfaces in S^4 for which the quantity S+λ2, where S is the squared norm of the second fundamental form and λ2 is the smaller eigenvalue of Lu's fundamental matrix, is constant. The main theorem states that the constant can only be 0 or 2: c=0 gives the totally geodesic 2-sphere, and c=2 gives either a Clifford torus in a totally geodesic S^3 or the Veronese surface in S^4. In particular, no closed minimal surface in S^4 has constant S+λ2>2, which verifies Lu's second-gap conjecture for (n,m)=(2,2). The proof splits into the range c≤2, handled by Lu's first-gap theorem, and c>2, which is excluded by a local moving-frame computation (Lemmas 3.1-3.2) and a maximum-principle argument, with degenerate cases λ2≡0 and λ1≡λ2 treated separately in Lemmas 4.2-4.3.

Significance. If the result holds, it completes the codimension-two case of Lu's second-gap conjecture for minimal surfaces, complementing the hypersurface result of Peng-Terng and the counterexamples of Li-Zhao in codimension at least three. The proof is transparent: the key algebraic identities (3.5), (3.9)-(3.11) are stated explicitly and are checkable; the use of Morrey's analyticity theorem in Lemma 4.1 is legitimate, and the maximum-principle contradiction is logically coherent. The paper also gives a clean contrast with higher codimensions, where counterexamples with dense constant values exist. The reliance on standard external results (Lu's first-gap theorem, Calabi-do Carmo-Wallach classification) is appropriate and does not introduce circularity.

minor comments (5)

[Lemma 4.2] After proving ω12=0, the text says 'and (2.2) yields K=0'. Substituting the shape operators into (2.2) gives K=1-κ^2, not K=0 directly; K=0 follows from dω12=-K dμ together with ω12=0. Please rephrase to avoid this misleading statement.
[Lemma 4.3] The sentence 'Choose the orthonormal normal frame such that (2.5) holds' conflicts with (2.5), where a>b>0 is assumed; in Lemma 4.3 one has a=b. This is a minor notation issue, but it should be clarified that an analogous normal form with equal entries is being used.
[Theorem 1.4] In item (iii), 'M is the Veronese surface in S^4' is slightly ambiguous if M is considered as an abstract domain: the standard Veronese immersion can be viewed either as S^2→S^4 or as its quotient RP^2→S^4. Consider stating that the image of F is the Veronese surface, or clarifying the intended convention.
[Proof of Theorem 1.4] The sentence 'Thus the conclusion of the theorem holds whenever c≤2' after the table is compressed: for 0<c<2 the table actually shows that no such immersion exists, since none of the listed cases has that value. Expanding this one line would improve readability.
[Lemma 4.1] The step from Morrey's local analyticity to a real-analytic structure on the connected surface M is standard but implicit. A short sentence explaining that the isothermal coordinate changes are real-analytic, or that one works on the analytic atlas generated by these charts, would remove potential concern about the identity theorem.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the proof is driven by local identities and external classification theorems; cited self-work is contextual.

full rationale

The derivation chain does not reduce to its inputs. Theorem 1.4 assumes S+λ2≡c as a hypothesis and derives c∈{0,2}; c is a quantified constant, not a parameter fitted to the conclusion. Lemmas 3.1–3.2 are differential identities following from Codazzi, Gauss, and Ricci equations together with the constancy hypothesis; no equation is defined in terms of the target classification. Lemma 4.1 invokes Morrey's analyticity theorem, an external classical result, only to apply the identity theorem; Lemma 4.3 uses the external Calabi–do Carmo–Wallach classification for the Veronese identification; the c≤2 case uses Lu's first-gap theorem, which is not the authors' own result. The self-citations that appear ([7], [10], [16], etc.) are contextual—Theorem 1.3 from [7] is not used in the proof, and [10]/[16] are background/counterexample discussion—so none is load-bearing. No step exhibits a fitted input renamed as a prediction, a definition in terms of the conclusion, or a uniqueness claim imported from the authors' prior work. The paper is self-contained against external benchmarks for its central claim.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters are fitted or chosen to force the result; the constant c is a quantified input. The proof relies on standard external theorems (Lu's first-gap equality case, Morrey analyticity, CDW classification, Gauss–Bonnet). No new physical or geometric entities are introduced.

assumptions (4)

domain assumption Lu's first-gap theorem (Theorem 1.1 of [17]): if 0≤S+λ2≤n on a closed minimal submanifold of S^{n+m}, then M is totally geodesic, a Clifford torus in S^{n+1}, or the Veronese surface in S^4.
Used in the proof of Theorem 1.4 to classify the c≤2 cases; external theorem assumed without reproof.
standard math Morrey's analytic regularity for analytic nonlinear elliptic systems (Theorem of [18]).
Lemma 4.1 uses it to promote the minimal immersion to real-analytic so the identity theorem applies to the discriminant Φ.
domain assumption Calabi–do Carmo–Wallach classification of linearly full minimal 2-spheres of constant curvature ([1,2]).
Lemma 4.3 uses it to identify the constant-curvature sphere in the λ1≡λ2 case with the Veronese surface.
standard math Gauss–Bonnet theorem.
In Lemma 4.3, constant positive curvature K=1/3 on the oriented double cover forces it to be a 2-sphere.

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Cite this review

Pith. "Pith review of Lu's conjecture for minimal surfaces in codimension two." pith.science (2026). https://pith.science/paper/TH24VJIF

@misc{pith2026260721336,
  author       = {Pith},
  title        = {Pith review of: Lu's conjecture for minimal surfaces in codimension two},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TH24VJIF}},
  note         = {Machine review of arXiv:2607.21336}
}

abstract

Let $M^2\to\mathbb{S}^4$ be a closed minimal immersion, let $S$ be the squared norm of its second fundamental form, and let $\lambda_1\geq\lambda_2\geq0$ be the eigenvalues of Lu's fundamental matrix. We classify all such immersions for which $S+\lambda_2$ is constant. We prove that the constant can only be $0$ or $2$. In the first case the image is a totally geodesic $2$-sphere; in the second case it is either a Clifford torus in a totally geodesic $\mathbb{S}^3$ or the Veronese surface in $\mathbb{S}^4$. In particular, there is no closed minimal surface in $\mathbb{S}^4$ with constant $S+\lambda_2>2$. Consequently, Lu's second-gap conjecture holds for minimal surfaces in codimension two. Together with the hypersurface result of Peng--Terng and the counterexamples of Li--Zhao in every codimension $m\geq3$, this completes the codimension picture for minimal surfaces.

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Works this paper leans on

23 extracted references · 5 linked inside Pith

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Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.

[1] [1]

Calabi,Minimal immersions of surfaces in Euclidean spheres, J

E. Calabi,Minimal immersions of surfaces in Euclidean spheres, J. Differential Geom.1(1967), 111–125

1967

[2] [2]

do Carmo and N

M. do Carmo and N. R. Wallach,Minimal immersions of spheres into spheres, Ann. of Math. (2) 93(1971), 43–62

1971

[3] [3]

S. P. Chang,On minimal hypersurfaces with constant scalar curvature in S4, J. Differential Geom. 37(1993), 523–534

1993

[4] [4]

S. S. Chern, M. do Carmo, and S. Kobayashi,Minimal submanifolds of a sphere with second fundamental form of constant length, in:Functional Analysis and Related Fields, Springer, New York, 1970, 59–75

1970

[5] [5]

P. J. De Smet, F. Dillen, L. Verstraelen, and L. Vrancken,A pointwise inequality in submanifold theory, Arch. Math. (Brno)35(1999), 115–128

1999

[6] [6]

Q. T. Deng and Y. J. Kou,Closed minimal hypersurfaces in S5(1)with constant scalar and Gauss–Kronecker curvatures, preprint, arXiv:2607.06588, 2026

arXiv 2026

[7] [7]

W. R. Ding, J. Q. Ge, F. G. Li, and X. Z. Yang,Lu’s conjecture for minimal surfaces, preprint, arXiv:2601.07194, 2026

arXiv 2026

[8] [8]

J. Q. Ge, F. G. Li, Z. Z. Tang, and Y. Zhou,A survey on the DDVV-type inequalities, Adv. Math. (China)53(2024), 449–467

2024