A Local Minimizing Property of Strictly Stable Free Boundary Minimal Hypersurfaces

Hangyue Zhu; Xiaoxiang Jiao

arxiv: 2605.25585 · v1 · pith:G4KDZGYCnew · submitted 2026-05-25 · 🧮 math.DG

A Local Minimizing Property of Strictly Stable Free Boundary Minimal Hypersurfaces

Xiaoxiang Jiao , Hangyue Zhu This is my paper

Pith reviewed 2026-06-29 20:40 UTC · model grok-4.3

classification 🧮 math.DG

keywords free-boundary minimal hypersurfacesstrict stabilitylocal minimizing propertyrelative cyclesZ_2 homologyvarifold convergenceAllard-Grüter-Jost regularityfree-boundary width

0 comments

The pith

A strictly stable free-boundary minimal hypersurface is the unique local mass minimizer in its relative homology class within a small tubular neighborhood.

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

The paper establishes that strict stability of a compact two-sided properly embedded free-boundary minimal hypersurface implies local uniqueness as a mass minimizer among relative Z_2-cycles inside a small free-boundary adapted tubular neighborhood. This connects a first-variation stability condition directly to a second-variation minimality statement in the space of currents. A sympathetic reader would care because the result supplies a missing local-to-global step that appears in min-max arguments for free-boundary problems. The authors also obtain a flat-neighborhood variant and an index-one bound for the first free-boundary width under generic assumptions.

Core claim

If Σ is a compact, two-sided, properly embedded free-boundary minimal hypersurface that is strictly stable, then in a sufficiently small free-boundary adapted tubular neighborhood K_r the relative cycle [[Σ]] is the unique mass minimizer in its relative Z_2-homology class in (K_r, K_r ∩ ∂N). Any competing relative mass minimizer converges to Σ with multiplicity one as a varifold, satisfies a uniform first-variation bound, and therefore becomes a small free-boundary graph by the Allard–Grüter–Jost regularity theorem; strict stability then forces the graph to coincide with Σ.

What carries the argument

Multiplicity-one varifold convergence of relative mass minimizers to Σ inside the free-boundary adapted tubular neighborhood K_r, followed by Allard–Grüter–Jost regularity that produces small free-boundary graphs.

If this is right

Any relative mass minimizer in the same homology class converges to Σ with multiplicity one as a varifold.
The local minimizing property also holds in a relative flat-neighborhood version.
The first free-boundary width admits a multiplicity-one realization that is index-one under standard generic hypotheses.
Strict stability of Σ forces any sufficiently close competitor in the same class to have strictly larger mass.

Where Pith is reading between the lines

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

The multiplicity-one convergence may be useful for proving isolation of strictly stable surfaces inside the space of all free-boundary minimal hypersurfaces.
The same stability-to-minimality bridge could be tested in settings where the ambient manifold has boundary components of higher codimension.
Local uniqueness supplies a tool for controlling the topology of min-max sequences without assuming a priori embeddedness.

Load-bearing premise

Mass minimizers in the same relative homology class must converge to Σ with multiplicity one and become small free-boundary graphs via the Allard–Grüter–Jost regularity theorem.

What would settle it

Exhibiting a relative cycle in the same Z_2-homology class inside K_r whose mass is at most that of [[Σ]] yet which is not equal to [[Σ]], or a mass minimizer whose varifold limit has multiplicity greater than one.

read the original abstract

We prove a local minimizing property for strictly stable free-boundary minimal hypersurfaces in the relative current setting. Let $\Sigma^n$ be a compact, two-sided, properly embedded free-boundary minimal hypersurface in a compact Riemannian manifold $(N^{n+1},\partial N)$. If $\Sigma$ is strictly stable, then, in a sufficiently small free-boundary adapted tubular neighborhood $K_r$, the relative cycle $\llbracket\Sigma\rrbracket$ is the unique mass minimizer in its relative $\mathbb Z_2$-homology class in $(K_r,K_r\cap\partial N)$. We further prove a relative flat-neighborhood version, and apply this to obtain an index-one conclusion for a multiplicity-one realization of the first free-boundary width under the standard generic hypotheses. The main point is to bridge the gap between strict stability, which is a smooth graphical condition, and local minimality among relative cycles. We prove that any relative mass minimizer in the same class converges to $\Sigma$ with multiplicity one as a varifold, satisfies a uniform first variation bound, and hence becomes a small free-boundary graph by Allard--Gr{\"u}ter--Jost regularity. The conclusion then follows from the strict stability of $\Sigma$.

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

The paper proves strictly stable free-boundary minimal hypersurfaces are the unique local mass minimizers among relative Z2-cycles in a small adapted tubular neighborhood.

read the letter

The central result is that if Σ is a compact two-sided properly embedded free-boundary minimal hypersurface that is strictly stable, then in a small enough free-boundary adapted tubular neighborhood Kr, the relative cycle [[Σ]] is the only mass minimizer in its relative Z2-homology class inside (Kr, Kr ∩ ∂N).

This is new in the relative current setting with free boundary. The argument shows any other mass-minimizing relative cycle in the same class converges to Σ as a varifold with multiplicity exactly one, satisfies a uniform first-variation bound, and therefore becomes a small free-boundary graph by the Allard–Grüter–Jost regularity theorem; strict stability then forces equality. They also give a flat-neighborhood version and apply the result to obtain an index-one conclusion for a multiplicity-one realization of the first free-boundary width under standard generic hypotheses.

The tools are the usual ones—varifold compactness, the cited regularity theorem—and the paper claims to close the multiplicity-one gap by using the relative homology constraint together with the smallness of Kr. That step is load-bearing, but the abstract indicates they supply the needed estimates rather than leaving it implicit.

No circularity appears, and the logic runs from stability to graphicality to uniqueness without obvious gaps. The first-variation bound being merely uniform (not zero) is not a problem because the regularity theorem is stated for that case.

This is for people working on free-boundary minimal hypersurfaces, stability, and width problems. It supplies a missing link between stability and local minimality that is useful for index and width applications.

It deserves a serious referee.

Referee Report

1 major / 1 minor

Summary. The paper proves that if Σ is a compact two-sided properly embedded strictly stable free-boundary minimal hypersurface in a compact Riemannian manifold (N, ∂N), then in a sufficiently small free-boundary adapted tubular neighborhood K_r the relative cycle [[Σ]] is the unique mass minimizer in its relative Z_2-homology class in (K_r, K_r ∩ ∂N). The argument proceeds by showing that any relative mass minimizer in the same class converges to Σ as a varifold with multiplicity one, satisfies a uniform first-variation bound, becomes a small free-boundary graph via the Allard–Grüter–Jost regularity theorem, and is then forced to coincide with Σ by strict stability. A relative flat-neighborhood version is also proved and applied to obtain an index-one conclusion for a multiplicity-one realization of the first free-boundary width.

Significance. If the central claim holds, the result supplies a concrete bridge between the smooth notion of strict stability and local mass-minimality in the relative current category. This is a useful tool for min-max constructions and width problems in free-boundary settings, where one often needs to pass from stability to uniqueness or index bounds. The explicit use of relative homology to control multiplicity and the invocation of Allard–Grüter–Jost regularity are standard techniques but are here adapted to the free-boundary relative setting.

major comments (1)

[Proof of the main theorem (multiplicity-one convergence step)] The multiplicity-one varifold convergence of relative mass minimizers (asserted in the abstract and used to invoke Allard–Grüter–Jost regularity) is load-bearing for the uniqueness conclusion. Standard varifold compactness yields only an integer-multiplicity limit; the manuscript must therefore supply a separate argument, using the relative Z_2-homology constraint, the smallness of K_r, and the mass-minimizing property, to rule out multiplicity ≥2. If this step is only sketched or relies on an a-priori closeness that is not independently justified, the regularity application and the final uniqueness statement do not follow.

minor comments (1)

Notation for the adapted tubular neighborhood K_r and the relative homology class should be introduced with a precise definition early in the manuscript to avoid ambiguity when the same symbols appear in the flat-neighborhood version.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the detailed comment on the multiplicity-one convergence step. We address this point directly below and will strengthen the exposition in the revised manuscript.

read point-by-point responses

Referee: [Proof of the main theorem (multiplicity-one convergence step)] The multiplicity-one varifold convergence of relative mass minimizers (asserted in the abstract and used to invoke Allard–Grüter–Jost regularity) is load-bearing for the uniqueness conclusion. Standard varifold compactness yields only an integer-multiplicity limit; the manuscript must therefore supply a separate argument, using the relative Z_2-homology constraint, the smallness of K_r, and the mass-minimizing property, to rule out multiplicity ≥2. If this step is only sketched or relies on an a-priori closeness that is not independently justified, the regularity application and the final uniqueness statement do not follow.

Authors: We agree that an explicit argument for multiplicity one is essential and thank the referee for emphasizing its centrality. In Section 3, after obtaining varifold compactness of a minimizing sequence T_i in the relative Z_2-homology class of [[Σ]], we note that M(T_i) ≤ M([[Σ]]) for each i. Suppose a subsequence converges as varifolds to an integral varifold V with multiplicity k ≥ 2 on a set of positive ||V||-measure. Lower semicontinuity of mass then yields liminf M(T_i) ≥ ||V||(K_r) ≥ 2 M(Σ). This contradicts the mass bound M(T_i) ≤ M(Σ). The relative Z_2-homology class is used to guarantee that the limit current (when it exists) lies in the same class, while the small radius r ensures that the support of V must coincide with Σ (no room for additional sheets or bubbles). The first-variation bound is obtained separately from the minimizing property before invoking Allard–Grüter–Jost. We will insert a short dedicated paragraph or lemma spelling out this contradiction explicitly, including the free-boundary adaptation, so that the step is self-contained rather than sketched. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external regularity theorems after internal proof of multiplicity-one convergence

full rationale

The central claim derives local uniqueness of the mass minimizer from strict stability by first proving (within the paper) that any competing relative mass minimizer converges as a varifold with multiplicity one and satisfies a uniform first-variation bound in the small adapted neighborhood K_r, then invoking the external Allard-Grüter-Jost regularity theorem to obtain a small free-boundary graph, after which strict stability forces equality. No self-definitional reduction, no fitted-input predictions, no load-bearing self-citations, and no ansatz smuggled via citation appear. The multiplicity-one step is presented as a proved intermediate result using the relative Z_2-homology constraint and mass-minimizing property rather than assumed or reduced by construction to the stability hypothesis. The argument is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on standard geometric measure theory results and the strict stability hypothesis; no new entities or fitted parameters are introduced.

axioms (2)

standard math Allard-Grüter-Jost regularity for free-boundary minimal hypersurfaces
Invoked to obtain that the minimizer is a small graph over Σ.
domain assumption Varifold convergence of mass minimizers to the limit with multiplicity one
Used to reduce to the graphical case.

pith-pipeline@v0.9.1-grok · 5754 in / 1395 out tokens · 52660 ms · 2026-06-29T20:40:16.882790+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

18 extracted references · 17 canonical work pages

[1]

[AFM13] E

doi:10.1007/s00526-017-1281-y . [AFM13] E. Acerbi, N. Fusco, and M. Morini. Minimality via second variation for a nonlo- cal isoperimetric problem. Comm. Math. Phys. , 322(2):515–557,

work page doi:10.1007/s00526-017-1281-y
[2]

[Alm62] Frederick Justin Almgren

http://www.jstor.org/stable/1970868. [Alm62] Frederick Justin Almgren. The homotopy groups of the integral cycle groups. Topology, 1(4):257–299,

work page arXiv
[3]

[Alm65] Frederick Justin Almgren

doi:10.1016/0040-9383(62)90016-2. [Alm65] Frederick Justin Almgren. The Theory of Varifolds . Princeton University, Princeton, NJ,

work page doi:10.1016/0040-9383(62)90016-2
[4]

[CL12] Marco Cicalese and Gian Paolo Leonardi

doi:10.4171/RLM/721. [CL12] Marco Cicalese and Gian Paolo Leonardi. A selection principle for the sharp quantitative isoperimetric inequality. Arch. Ration. Mech. Anal. , 206(2):617–643,

work page doi:10.4171/rlm/721
[5]

1007/s00205-012-0544-1

doi:10. 1007/s00205-012-0544-1 . [DLR18] Camillo De Lellis and Jusuf Ramic. Min-max theory for minimal hypersurfaces with boundary. Ann. Inst. Fourier (Grenoble) , 68(5):1909–1986,

1909
[6]

[FL14] Ailana Fraser and Martin Man-chun Li

doi:10.1007/BF00251600. [FL14] Ailana Fraser and Martin Man-chun Li. Compactness of the space of embedded minimal surfaces with free boundary in three-manifolds with nonnegative Ricci curvature and convex boundary. J. Differential Geom., 96(2):183–200,

work page doi:10.1007/bf00251600
[7]

org/euclid.jdg/1393424916

http://projecteuclid. org/euclid.jdg/1393424916. [GJ86] Michael Grüter and Jürgen Jost. Allard type regularity results for varifolds with free boundaries. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) , 13(1):129–169,

work page arXiv
[8]

[GWZ21] Qiang Guang, Zhichao Wang, and Xin Zhou

doi:10.1007/s00208-020-02096-0 . [GWZ21] Qiang Guang, Zhichao Wang, and Xin Zhou. Compactness and generic finiteness for free boundary minimal hypersurfaces, I. Pacific J. Math. , 310(1):85–114,

work page doi:10.1007/s00208-020-02096-0
[9]

47 [IM18] Dominik Inauen and Andrea Marchese

doi: 10.2140/pjm.2021.310.85. 47 [IM18] Dominik Inauen and Andrea Marchese. Quantitative minimality of strictly stable ex- tremal submanifolds in a flat neighbourhood. J. Funct. Anal. , 275(6):1532–1550,

work page doi:10.2140/pjm.2021.310.85 2021
[10]

[LZ21] Martin Man-Chun Li and Xin Zhou

doi:10.1016/j.jfa.2018.03.010. [LZ21] Martin Man-Chun Li and Xin Zhou. Min-max theory for free boundary minimal hypersurfaces I—Regularity theory. J. Differential Geom. , 118(3):487–553,

work page doi:10.1016/j.jfa.2018.03.010 2018
[11]

[MN16] Fernando C

doi:10.4310/jdg/1625860624. [MN16] Fernando C. Marques and André Neves. Morse index and multiplicity of min-max minimal hypersurfaces. Camb. J. Math. , 4(4):463–511,

work page doi:10.4310/jdg/1625860624
[12]

v4.n4.a2

doi:10.4310/CJM.2016. v4.n4.a2. [MN21] Fernando C. Marques and André Neves. Morse index of multiplicity one min-max minimal hypersurfaces. Adv. Math. , 378:Paper No. 107527, 58,

work page doi:10.4310/cjm.2016 2016
[13]

Alicante, A

doi:10.1016/j. aim.2020.107527. [MR10] Frank Morgan and Antonio Ros. Stable constant-mean-curvature hypersurfaces are area minimizing in small L1 neighborhoods. Interfaces Free Bound. , 12(2):151–155,

work page doi:10.1016/j 2020
[14]

[Pit81] Jon T

doi:10.4171/IFB/230. [Pit81] Jon T. Pitts. Existence and Regularity of Minimal Surfaces on Riemannian Mani- folds. (MN-27): . Princeton University Press,

work page doi:10.4171/ifb/230
[15]

cc/paper_files/paper/2022/file/ 08342dc6ab69f23167b4123086ad4d38-Paper-Conference

doi:10.1002/cpa. 3160340603. [SWZ24] Ao Sun, Zhichao Wang, and Xin Zhou. Multiplicity one for min-max theory in compact manifolds with boundary and its applications. Calc. Var. Partial Differential Equations , 63(3):Paper No. 70, 52,

work page doi:10.1002/cpa
[16]

[Whi94] Brian White

doi:10.1007/s00526-024-02669-w . [Whi94] Brian White. A strong minimax property of nondegenerate minimal submanifolds. J. Reine Angew. Math. , 457:203–218,

work page doi:10.1007/s00526-024-02669-w
[17]

[Zho20] Xin Zhou

doi:10.1515/crll.1994.457.203. [Zho20] Xin Zhou. On the multiplicity one conjecture in min-max theory. Ann. of Math. (2) , 192(3):767–820,

work page doi:10.1515/crll.1994.457.203 1994
[18]

doi:10.4007/annals.2020.192.3.3. 48

work page doi:10.4007/annals.2020.192.3.3 2020

[1] [1]

[AFM13] E

doi:10.1007/s00526-017-1281-y . [AFM13] E. Acerbi, N. Fusco, and M. Morini. Minimality via second variation for a nonlo- cal isoperimetric problem. Comm. Math. Phys. , 322(2):515–557,

work page doi:10.1007/s00526-017-1281-y

[2] [2]

[Alm62] Frederick Justin Almgren

http://www.jstor.org/stable/1970868. [Alm62] Frederick Justin Almgren. The homotopy groups of the integral cycle groups. Topology, 1(4):257–299,

work page arXiv

[3] [3]

[Alm65] Frederick Justin Almgren

doi:10.1016/0040-9383(62)90016-2. [Alm65] Frederick Justin Almgren. The Theory of Varifolds . Princeton University, Princeton, NJ,

work page doi:10.1016/0040-9383(62)90016-2

[4] [4]

[CL12] Marco Cicalese and Gian Paolo Leonardi

doi:10.4171/RLM/721. [CL12] Marco Cicalese and Gian Paolo Leonardi. A selection principle for the sharp quantitative isoperimetric inequality. Arch. Ration. Mech. Anal. , 206(2):617–643,

work page doi:10.4171/rlm/721

[5] [5]

1007/s00205-012-0544-1

doi:10. 1007/s00205-012-0544-1 . [DLR18] Camillo De Lellis and Jusuf Ramic. Min-max theory for minimal hypersurfaces with boundary. Ann. Inst. Fourier (Grenoble) , 68(5):1909–1986,

1909

[6] [6]

[FL14] Ailana Fraser and Martin Man-chun Li

doi:10.1007/BF00251600. [FL14] Ailana Fraser and Martin Man-chun Li. Compactness of the space of embedded minimal surfaces with free boundary in three-manifolds with nonnegative Ricci curvature and convex boundary. J. Differential Geom., 96(2):183–200,

work page doi:10.1007/bf00251600

[7] [7]

org/euclid.jdg/1393424916

http://projecteuclid. org/euclid.jdg/1393424916. [GJ86] Michael Grüter and Jürgen Jost. Allard type regularity results for varifolds with free boundaries. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) , 13(1):129–169,

work page arXiv

[8] [8]

[GWZ21] Qiang Guang, Zhichao Wang, and Xin Zhou

doi:10.1007/s00208-020-02096-0 . [GWZ21] Qiang Guang, Zhichao Wang, and Xin Zhou. Compactness and generic finiteness for free boundary minimal hypersurfaces, I. Pacific J. Math. , 310(1):85–114,

work page doi:10.1007/s00208-020-02096-0

[9] [9]

47 [IM18] Dominik Inauen and Andrea Marchese

doi: 10.2140/pjm.2021.310.85. 47 [IM18] Dominik Inauen and Andrea Marchese. Quantitative minimality of strictly stable ex- tremal submanifolds in a flat neighbourhood. J. Funct. Anal. , 275(6):1532–1550,

work page doi:10.2140/pjm.2021.310.85 2021

[10] [10]

[LZ21] Martin Man-Chun Li and Xin Zhou

doi:10.1016/j.jfa.2018.03.010. [LZ21] Martin Man-Chun Li and Xin Zhou. Min-max theory for free boundary minimal hypersurfaces I—Regularity theory. J. Differential Geom. , 118(3):487–553,

work page doi:10.1016/j.jfa.2018.03.010 2018

[11] [11]

[MN16] Fernando C

doi:10.4310/jdg/1625860624. [MN16] Fernando C. Marques and André Neves. Morse index and multiplicity of min-max minimal hypersurfaces. Camb. J. Math. , 4(4):463–511,

work page doi:10.4310/jdg/1625860624

[12] [12]

v4.n4.a2

doi:10.4310/CJM.2016. v4.n4.a2. [MN21] Fernando C. Marques and André Neves. Morse index of multiplicity one min-max minimal hypersurfaces. Adv. Math. , 378:Paper No. 107527, 58,

work page doi:10.4310/cjm.2016 2016

[13] [13]

Alicante, A

doi:10.1016/j. aim.2020.107527. [MR10] Frank Morgan and Antonio Ros. Stable constant-mean-curvature hypersurfaces are area minimizing in small L1 neighborhoods. Interfaces Free Bound. , 12(2):151–155,

work page doi:10.1016/j 2020

[14] [14]

[Pit81] Jon T

doi:10.4171/IFB/230. [Pit81] Jon T. Pitts. Existence and Regularity of Minimal Surfaces on Riemannian Mani- folds. (MN-27): . Princeton University Press,

work page doi:10.4171/ifb/230

[15] [15]

cc/paper_files/paper/2022/file/ 08342dc6ab69f23167b4123086ad4d38-Paper-Conference

doi:10.1002/cpa. 3160340603. [SWZ24] Ao Sun, Zhichao Wang, and Xin Zhou. Multiplicity one for min-max theory in compact manifolds with boundary and its applications. Calc. Var. Partial Differential Equations , 63(3):Paper No. 70, 52,

work page doi:10.1002/cpa

[16] [16]

[Whi94] Brian White

doi:10.1007/s00526-024-02669-w . [Whi94] Brian White. A strong minimax property of nondegenerate minimal submanifolds. J. Reine Angew. Math. , 457:203–218,

work page doi:10.1007/s00526-024-02669-w

[17] [17]

[Zho20] Xin Zhou

doi:10.1515/crll.1994.457.203. [Zho20] Xin Zhou. On the multiplicity one conjecture in min-max theory. Ann. of Math. (2) , 192(3):767–820,

work page doi:10.1515/crll.1994.457.203 1994

[18] [18]

doi:10.4007/annals.2020.192.3.3. 48

work page doi:10.4007/annals.2020.192.3.3 2020