Free boundary minimal annuli in $S^2_+\times S^1$

Juncheol Pyo; Keomkyo Seo; Pak Tung Ho

arxiv: 2505.10826 · v1 · submitted 2025-05-16 · 🧮 math.DG

Free boundary minimal annuli in S²_+times S¹

Pak Tung Ho , Juncheol Pyo , Keomkyo Seo This is my paper

Pith reviewed 2026-05-22 15:29 UTC · model grok-4.3

classification 🧮 math.DG

keywords free boundary minimal surfacescompactnessstrict convexityannulinonnegative Ricci curvature3-manifoldscounterexample

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The pith

Strict convexity of the boundary is necessary for compactness of free boundary minimal annuli in 3-manifolds with nonnegative Ricci curvature.

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

Fraser and Li proved compactness for compact properly embedded free boundary minimal surfaces of fixed topology in compact 3-manifolds with nonnegative Ricci curvature when the boundary is strictly convex. This paper shows the strict convexity condition cannot be relaxed. It does so by producing a sequence of free boundary minimal annuli in the manifold S²₊ × S¹ that fails to be compact, even though the boundary there is convex but not strictly convex. A reader cares because the example identifies the precise boundary condition needed to prevent degeneration or loss of compactness in these geometric settings.

Core claim

The authors exhibit a sequence of free boundary minimal annuli in the compact 3-manifold S²₊ × S¹ with nonnegative Ricci curvature whose boundary is convex but not strictly convex, and this sequence fails to remain compact, establishing that the strict convexity assumption in prior compactness theorems is necessary.

What carries the argument

The manifold S²₊ × S¹ with its product metric, serving as the ambient space that admits a non-compact sequence of free boundary minimal annuli.

If this is right

Compactness theorems for free boundary minimal surfaces of annular type require the boundary to be strictly convex.
In the presence of a convex but not strictly convex boundary, sequences of free boundary minimal annuli can escape compactness while remaining embedded.
The counterexample is specific to the topology of annuli in this particular manifold.
The result indicates that nonnegative Ricci curvature alone is insufficient to guarantee compactness without the strict convexity hypothesis.

Where Pith is reading between the lines

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

The example suggests that similar counterexamples could be constructed for other convex but non-strictly convex boundaries in 3-manifolds.
It raises the question of whether a weaker notion of convexity, such as mean-convexity, suffices for compactness in special cases.
The construction may connect to degeneration phenomena in other free boundary problems where boundary curvature controls the behavior of minimal surfaces.

Load-bearing premise

There exists a sequence of distinct free boundary minimal annuli in a compact 3-manifold with nonnegative Ricci curvature and convex but not strictly convex boundary that does not converge to a compact limit surface.

What would settle it

A direct computation or construction showing that every sequence of free boundary minimal annuli in S²₊ × S¹ converges smoothly to a compact free boundary minimal annulus would falsify the claim.

read the original abstract

Let $M$ be a compact 3-dimensional Riemannian manifold with nonnegative Ricci curvature and a nonempty boundary $\partial M$. Fraser and Li \cite{Fraser&Li} established a compactness theorem for the space of compact, properly embedded minimal surfaces of fixed topological type in $M$ with a free boundary on $\partial M$, assuming that $\partial M$ is strictly convex with respect to the inward unit normal. In this paper, we show that the strict convexity condition on $\partial M$ cannot be relaxed.

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

This paper gives a concrete counterexample in a product manifold showing that strict convexity of the boundary cannot be relaxed in the Fraser-Li compactness theorem for free boundary minimal surfaces.

read the letter

Colleague, the key point is that this work establishes sharpness for the strict convexity hypothesis in Fraser and Li's theorem on compactness of free boundary minimal surfaces in compact 3-manifolds with nonnegative Ricci curvature. The authors construct a sequence of embedded free boundary minimal annuli in the product of the closed upper hemisphere with a circle whose second fundamental form norms become unbounded, even though the boundary is totally geodesic and thus convex but not strictly convex. This directly shows the condition is necessary rather than just convenient. What the paper does well is pick a simple symmetric setting where explicit constructions are feasible. The product structure lets them produce the annuli by varying a parameter in a controlled way, keeping the surfaces embedded and satisfying the free boundary condition while the curvature blows up. That makes the failure of compactness visible without heavy machinery. The Ricci curvature bound holds in this example, and the boundary geometry matches what the theorem requires except for the strictness. I see no load-bearing gaps in the argument from the details given; the sequence appears to be properly constructed and the deduction that compactness fails follows cleanly. Minor points could include checking that the annuli stay minimal at every stage of the degeneration or confirming the embedding property in the limit, but these look handled. This is for people working on free boundary minimal surfaces or compactness results in geometric analysis. A reader who cites or applies the Fraser-Li theorem would want to know this sharpness result to state their own theorems accurately. The paper shows clear engagement with the literature and produces a verifiable counterexample, so it deserves a serious referee even if the result is narrow. I would recommend sending it out for peer review.

Referee Report

0 major / 3 minor

Summary. The paper constructs an explicit counterexample showing that the strict convexity assumption on the boundary cannot be relaxed in the compactness theorem of Fraser and Li for properly embedded free boundary minimal surfaces of fixed topology in compact 3-manifolds with nonnegative Ricci curvature. In the product manifold S²₊ × S¹ equipped with the product metric, the boundary is convex but not strictly convex; the authors produce a sequence of embedded free boundary minimal annuli whose second fundamental form norms become unbounded, so the space fails to be compact.

Significance. The result is significant because it establishes sharpness of the strict-convexity hypothesis in an important compactness theorem. The construction is carried out in a simple, geometrically natural manifold with Ric(M) ≥ 0 and totally geodesic boundary, and the sequence is embedded, which strengthens the counterexample. The work supplies a concrete, falsifiable instance rather than an abstract existence argument.

minor comments (3)

[Section 2] In the definition of the product metric on S²₊ × S¹, explicitly state the range of the S¹ coordinate and confirm that the boundary components are totally geodesic with respect to the inward normal.
[Section 3] The statement that the annuli are free-boundary minimal would benefit from a brief verification that the conormal condition holds along the two boundary circles.
[Section 4] Add a short remark comparing the curvature blow-up rate obtained here with the rate that would be forbidden under strict convexity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive report, which accurately summarizes the main result of the paper and highlights its significance in establishing the sharpness of the strict-convexity hypothesis in the Fraser-Li compactness theorem. We appreciate the recommendation to accept.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper constructs an explicit counterexample in the product manifold S^2_+ × S^1 (with totally geodesic boundary) to demonstrate that the strict convexity hypothesis in the cited Fraser-Li compactness theorem cannot be relaxed. The argument proceeds by direct verification that Ric(M) ≥ 0, that the boundary is convex but not strictly convex, and that a sequence of embedded free-boundary minimal annuli has unbounded second fundamental form. No step reduces a claimed prediction or uniqueness result to a fitted parameter, self-definition, or load-bearing self-citation; the central claim rests on an independent geometric construction rather than any internal loop.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard Riemannian geometry and minimal surface theory without introducing new free parameters or invented entities; the counterexample is constructed within existing axioms.

axioms (2)

standard math Standard theory of minimal surfaces with free boundary in Riemannian manifolds with boundary
Invoked throughout to define the objects and compactness notion.
domain assumption Nonnegative Ricci curvature implies certain curvature controls on minimal surfaces
Used to set up the ambient manifold class as in Fraser-Li.

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discussion (0)

Lean theorems connected to this paper

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

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1.2. … there exists a sequence of compact properly embedded minimal annuli … which is not compact. … the general solutions to (2.4) and (2.7) are elliptic functions for a sequence of initial values r(t=0)=c₁>c₂>…→0.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.