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arxiv: 2605.21607 · v1 · pith:AE42YGP7new · submitted 2026-05-20 · 🧮 math.DG · math.MG

Minimal spheres and scalar curvature

Talant Talipov This is my paper

Pith reviewed 2026-05-22 08:26 UTC · model grok-4.3

classification 🧮 math.DG math.MG
keywords minimal spheresscalar curvatureRicci curvaturethree-sphereembedded minimal surfacesYau conjectureellipsoids
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The pith

Any three-sphere with positive Ricci curvature and scalar curvature bounded below by a positive constant contains four distinct embedded minimal two-spheres with explicit area bounds.

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

The paper establishes a quantitative version of the Wang-Zhou theorem on minimal spheres in three-spheres. It assumes the metric has positive Ricci curvature together with scalar curvature at least some fixed positive number Lambda zero, then shows there must exist four distinct embedded minimal two-spheres whose areas satisfy explicit upper bounds of the form twelve pi times i plus one over Lambda zero. This supplies concrete size control rather than mere existence. The same curvature assumptions are then used to prove that ellipsoids in four-space with one sufficiently large semi-axis contain at least three non-planar embedded minimal two-spheres.

Core claim

Suppose that (S^3, g) has positive Ricci curvature and scalar curvature R_g greater than or equal to Lambda_0 greater than zero. Then there exist four distinct embedded minimal two-spheres Sigma one through four inside (S^3, g) such that the g-area of Sigma i is at most twelve pi times (i plus one) divided by Lambda_0 for each i from one to four. The same result is applied to show that ellipsoids centered at the origin in R^4 with one sufficiently large semi-axis contain at least three non-planar embedded minimal two-spheres.

What carries the argument

Quantitative version of the Wang-Zhou theorem, which uses positive Ricci curvature and a uniform positive lower bound on scalar curvature to guarantee existence and produce the stated area upper bounds for the minimal spheres.

If this is right

  • Under the stated curvature conditions the four minimal spheres exist and each satisfies the explicit area upper bound.
  • Ellipsoids with one sufficiently large semi-axis contain at least three non-planar embedded minimal two-spheres.
  • The area of the smallest such minimal sphere is at most twenty-four pi over Lambda zero.
  • The area bounds become sharper when the lower bound on scalar curvature is larger.

Where Pith is reading between the lines

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

  • The same curvature hypotheses might be weakened to Ricci curvature bounded below by a small positive constant if a corresponding quantitative estimate can be obtained.
  • The area-control technique could be tested on rotationally symmetric metrics to see how close the bounds come to being achieved.
  • Results of this type may help bound the number or size of minimal surfaces in other three-manifolds that satisfy similar curvature assumptions.

Load-bearing premise

The metric on the three-sphere must have positive Ricci curvature at every point and a uniform positive lower bound on its scalar curvature.

What would settle it

A Riemannian metric on the three-sphere with positive Ricci curvature and scalar curvature everywhere at least some fixed positive number Lambda zero, yet possessing fewer than four distinct embedded minimal two-spheres, would contradict the claim.

read the original abstract

In 1982, S.-T. Yau conjectured that there exist four distinct embedded minimal two-spheres in any manifold diffeomorphic to $S^3$. Wang-Zhou confirmed this conjecture for Riemannian three-spheres when the metric is bumpy or has positive Ricci curvature. We prove the following quantitative version of their theorem. Suppose that $(S^3,g)$ has positive Ricci curvature and scalar curvature $R_g\ge \Lambda_0>0$. Then there exist four distinct embedded minimal two-spheres $\Sigma_1,\ldots,\Sigma_4\subset (S^3,g)$ such that $\operatorname{area}_{g}(\Sigma_i)\le 12\pi(i+1)/\Lambda_0$ for every $i=1,\ldots,4$. We apply this result to a problem posed by S.-T. Yau in 1987 on whether the planar two-spheres are the only minimal spheres in ellipsoids centered at the origin in $\mathbb R^4$. Haslhofer-Ketover proved that ellipsoids with one sufficiently large semi-axis contain at least one non-planar embedded minimal two-sphere. We prove that such ellipsoids contain at least three non-planar embedded minimal two-spheres.

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

1 major / 2 minor

Summary. The manuscript proves a quantitative strengthening of the Wang-Zhou theorem: if (S^3, g) has positive Ricci curvature and scalar curvature R_g ≥ Λ_0 > 0, then there exist four distinct embedded minimal two-spheres Σ_i with area_g(Σ_i) ≤ 12π(i+1)/Λ_0 for i = 1,…,4. The result is applied to show that ellipsoids in R^4 with one sufficiently large semi-axis contain at least three non-planar embedded minimal two-spheres, extending the earlier existence result of Haslhofer-Ketover.

Significance. The quantitative area bounds controlled by the scalar-curvature lower bound constitute a useful refinement of the existence theorem and directly enable the multiplicity statement for non-planar spheres in ellipsoids. The argument relies on a quantitative version of the Wang-Zhou min-max construction together with standard curvature assumptions that are already present in the literature; the explicit constants make the result falsifiable and potentially applicable to further width estimates.

major comments (1)
  1. [§3] §3, proof of Theorem 1.1: the derivation of the precise factor (i+1) in the area bound 12π(i+1)/Λ_0 from the scalar-curvature assumption and the index of the min-max class should be spelled out explicitly; the current sketch leaves unclear whether the bound follows from a direct integration of the stability inequality or requires an additional covering argument.
minor comments (2)
  1. [Introduction] Introduction, paragraph 2: the citation to Yau’s 1987 problem would benefit from a one-sentence reminder that the planar spheres are the obvious minimal spheres in the round case.
  2. [§4] §4, application to ellipsoids: verify that the perturbed metrics used to apply the main theorem indeed satisfy positive Ricci curvature uniformly; a short computation or reference to the second-variation formula would suffice.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript, the positive assessment of its significance, and the recommendation for minor revision. We address the single major comment below.

read point-by-point responses

  1. Referee: [§3] §3, proof of Theorem 1.1: the derivation of the precise factor (i+1) in the area bound 12π(i+1)/Λ_0 from the scalar-curvature assumption and the index of the min-max class should be spelled out explicitly; the current sketch leaves unclear whether the bound follows from a direct integration of the stability inequality or requires an additional covering argument.

    Authors: We agree that the sketch in the proof of Theorem 1.1 is too concise on this point. In the revised manuscript we will expand the relevant paragraph in §3 to give a fully explicit derivation. The factor (i+1) arises directly from integrating the stability inequality for the min-max surface against the coordinate functions furnished by the min-max class of index at most i, combined with the lower bound R_g ≥ Λ_0 and the positive Ricci curvature (which controls the second fundamental form via the Gauss equation). The resulting area estimate is obtained by a straightforward integration by parts and does not invoke any covering argument. We will write out the calculation line by line, including the precise use of the index bound, so that the origin of the constant 12π(i+1)/Λ_0 is transparent. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper establishes a quantitative version of the Wang-Zhou theorem for the existence of four embedded minimal two-spheres with explicit area bounds controlled by the lower bound on scalar curvature, under the assumption of positive Ricci curvature. This relies on cited external results (Wang-Zhou) and standard min-max techniques for minimal surfaces, without any reduction of the claimed area bounds or existence statements to quantities defined by the authors' own prior work or to fitted parameters presented as predictions. The application to the ellipsoid problem likewise extends independent prior results (Haslhofer-Ketover) using the new quantitative statement. No self-definitional, self-citation load-bearing, or ansatz-smuggling steps appear in the derivation chain; the argument remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on standard background results in geometric analysis (existence and regularity of minimal surfaces, index estimates) plus the cited Wang-Zhou theorem; no new free parameters, invented entities, or ad-hoc axioms are introduced in the abstract.

axioms (1)
  • standard math Standard existence and regularity theorems for minimal surfaces in Riemannian 3-manifolds with positive Ricci curvature
    Invoked to guarantee the four distinct embedded minimal 2-spheres exist under the stated curvature assumptions.

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

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    Relation between the paper passage and the cited Recognition theorem.

    Suppose that (S^3,g) has positive Ricci curvature and scalar curvature R_g ≥ Λ_0 > 0. Then there exist four distinct embedded minimal two-spheres Σ_1, …, Σ_4 ⊂ (S^3,g) such that area_g(Σ_i) ≤ 12π(i+1)/Λ_0

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Reference graph

Works this paper leans on

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