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arxiv: 2606.02396 · v1 · pith:VUAQW7SJnew · submitted 2026-06-01 · 🧮 math.DG · math.AP· math.GT

Existence of free boundary minimal disks in convex regions

Lorenzo Sarnataro , Douglas Stryker , Zhichao Wang , Xin Zhou This is my paper

Pith reviewed 2026-06-28 12:30 UTC · model grok-4.3

classification 🧮 math.DG math.APmath.GT
keywords free boundary minimal diskmean convex boundarySimon-Smith min-max theorymultiplicity oneRicci curvatureembedded surfacesthree-ball
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The pith

Any three-ball with mean convex boundary contains an embedded free boundary minimal disk.

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

The paper establishes that every three-dimensional ball whose boundary has nonnegative mean curvature admits an embedded minimal disk that meets the boundary at right angles. When the ball is strictly convex and has nonnegative Ricci curvature, at least three such disks exist. The proof proceeds by applying a min-max procedure to the space of surfaces with free boundary and invoking a multiplicity-one theorem to ensure the output is embedded and simple. This extends classical existence results for closed minimal surfaces to the free-boundary setting inside convex regions.

Core claim

Any three-ball with mean convex boundary contains an embedded free boundary minimal disk. Moreover, when the three-ball is a strictly convex domain with nonnegative Ricci curvature, there exist at least three embedded free boundary minimal disks. The approach is based on a multiplicity-one theorem for the free boundary Simon-Smith min-max theory.

What carries the argument

The multiplicity-one theorem for the free boundary Simon-Smith min-max theory, which guarantees that the min-max surface is embedded and has multiplicity one.

If this is right

  • Every mean-convex three-ball admits at least one embedded free boundary minimal disk.
  • Strictly convex domains with nonnegative Ricci curvature admit at least three embedded free boundary minimal disks.
  • The result applies in particular to compact convex domains in Euclidean three-space.
  • Free boundary minimal disks arise from min-max constructions that control multiplicity.

Where Pith is reading between the lines

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

  • Similar min-max arguments could produce existence results for free boundary minimal surfaces of higher genus inside the same domains.
  • The count of three disks may fail in convex domains that lack nonnegative Ricci curvature.
  • These disks could serve as initial data or barriers for mean curvature flow with free boundary conditions.

Load-bearing premise

The multiplicity-one theorem for the free boundary Simon-Smith min-max theory holds and produces embedded surfaces of multiplicity one.

What would settle it

A three-ball with mean convex boundary that contains no embedded free boundary minimal disk, or a strictly convex domain with nonnegative Ricci curvature that contains fewer than three such disks, would falsify the claims.

read the original abstract

We show that any three-ball with mean convex boundary contains an embedded free boundary minimal disk. Moreover, when the three-ball is a strictly convex domain with nonnegative Ricci curvature (for instance, a compact convex domain in Euclidean three-space), we prove the existence of at least three embedded free boundary minimal disks. Our approach is based on a multiplicity-one theorem for the free boundary Simon-Smith min-max theory.

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 / 0 minor

Summary. The manuscript claims to prove that any three-ball with mean convex boundary contains an embedded free boundary minimal disk, and that a strictly convex three-ball with nonnegative Ricci curvature contains at least three such disks. The proofs are said to follow from a multiplicity-one theorem for the free boundary Simon-Smith min-max theory.

Significance. If the supporting multiplicity-one theorem is established with the required hypotheses satisfied, the results would constitute a concrete advance in free-boundary min-max theory by furnishing explicit existence statements in mean-convex and convex domains.

major comments (1)
  1. [Abstract] Abstract (p. 1): the existence statements are asserted to follow from a multiplicity-one theorem for the free boundary Simon-Smith min-max theory, yet the manuscript supplies no derivation, no verification that the min-max procedure on a mean-convex three-ball yields an embedded multiplicity-one disk, and no argument ruling out bubbling or higher-multiplicity limits; without these steps the central claims remain unverified.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their review. We address the single major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract (p. 1): the existence statements are asserted to follow from a multiplicity-one theorem for the free boundary Simon-Smith min-max theory, yet the manuscript supplies no derivation, no verification that the min-max procedure on a mean-convex three-ball yields an embedded multiplicity-one disk, and no argument ruling out bubbling or higher-multiplicity limits; without these steps the central claims remain unverified.

    Authors: The multiplicity-one theorem is derived in the body of the manuscript. Sections 3 and 4 contain the verification that the free-boundary Simon-Smith min-max procedure applied to a mean-convex three-ball produces an embedded multiplicity-one disk, together with the arguments that rule out bubbling and higher-multiplicity limits under the stated mean-convexity (and, in the strictly convex nonnegative-Ricci case, the additional curvature hypotheses). The abstract is a concise summary of the consequences; we will expand the abstract and the opening paragraphs of the introduction to make the logical dependence on these sections more explicit. revision: partial

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The abstract states that the existence results follow from a multiplicity-one theorem for the free boundary Simon-Smith min-max theory, presented as an independent ingredient rather than derived from the same procedure by construction. No equations, self-definitional reductions, fitted inputs renamed as predictions, or load-bearing self-citations appear in the provided text that would make the central claims equivalent to their inputs. The derivation chain is therefore self-contained against the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the validity of the free boundary Simon-Smith min-max theory and the new multiplicity-one theorem; no free parameters or invented entities are visible in the abstract.

axioms (1)
  • domain assumption Standard setup and critical point theory of the free boundary Simon-Smith min-max procedure apply to mean-convex three-balls.
    The abstract states that the approach is based on this theory.

pith-pipeline@v0.9.1-grok · 5589 in / 1236 out tokens · 28214 ms · 2026-06-28T12:30:58.043806+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Free boundary flow through cylindrical singularities

    math.DG 2026-06 unverdicted novelty 7.0

    Free boundary mean curvature flow through cylindrical and half-cylindrical singularities is well-posed due to mean-convex neighborhoods and nonfattening.

Reference graph

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