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arxiv: 2604.13422 · v1 · submitted 2026-04-15 · 🧮 math.DG

Infinite existence of equivariant minimal hypersurfaces

Xingzhe Li , Tongrui Wang This is my paper

Pith reviewed 2026-05-10 12:55 UTC · model grok-4.3

classification 🧮 math.DG
keywords equivariant minimal hypersurfacesmin-max theoryRiemannian manifoldsLie group actionsG-homology classesmaximal cuttingsembedded minimal surfaces
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The pith

Closed Riemannian manifolds with isometric compact Lie group actions contain infinitely many G-invariant minimal hypersurfaces.

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

The paper proves that any closed Riemannian manifold M with an isometric action by a compact Lie group G must contain infinitely many distinct G-invariant minimal hypersurfaces. This follows from a new construction that builds min-max families while respecting the symmetry. The authors also prove a stronger statement under an extra finiteness condition: each G-homology class contains infinitely many distinct embedded minimal G-hypersurfaces. To carry out the argument they develop an equivariant version of min-max theory that works even in manifolds with cylindrical ends.

Core claim

In a closed Riemannian manifold M with an isometric action of a compact Lie group G, there exist infinitely many G-invariant minimal hypersurfaces. Under the additional assumption that M has only finitely many minimal G-hypersurfaces without a G-invariant unit normal, every G-homology class is realized by infinitely many distinct embedded minimal G-hypersurfaces. The construction proceeds via a new algorithm of multi-stage maximal cuttings inside an equivariant min-max procedure, which is also established for manifolds with cylindrical ends.

What carries the argument

multi-stage maximal cuttings algorithm that generates equivariant min-max families of G-invariant hypersurfaces

If this is right

  • Infinitely many G-invariant minimal hypersurfaces exist in every such symmetric closed manifold.
  • Under the finiteness assumption, each G-homology class admits infinitely many distinct embedded realizations by minimal G-hypersurfaces.
  • Equivariant min-max theory extends to manifolds with cylindrical ends.
  • The multi-stage cutting procedure produces families that remain invariant under the group action at every stage.

Where Pith is reading between the lines

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

  • Symmetry constraints imposed by the group action do not reduce the total number of minimal hypersurfaces.
  • The result may apply to studying minimal hypersurfaces on orbifolds obtained by quotienting by G.
  • The cylindrical-end min-max theory could be tested on non-compact symmetric spaces with controlled ends.
  • Concrete examples such as spheres with rotational symmetry groups could be checked to see whether the infinite families are visible by direct computation.

Load-bearing premise

M contains at most a finite number of minimal G-hypersurfaces admitting no G-invariant unit normal.

What would settle it

An explicit enumeration or construction that finds only finitely many distinct G-invariant minimal hypersurfaces in a closed manifold equipped with a nontrivial isometric compact Lie group action would disprove the claim.

read the original abstract

For a closed Riemannian manifold $M$ with a compact Lie group $G$ acting by isometries, we show that there are infinitely many $G$-invariant minimal hypersurfaces. Under the assumption that $M$ contains at most a finite number of minimal $G$-hypersurfaces admitting no $G$-invariant unit normal, we further show that each $G$-homology class of $M$ admits infinitely many distinct realizations by embedded minimal $G$-hypersurfaces. The proof relies on a new algorithm that employs multi-stage maximal cuttings. As part of this work, we also established an equivariant min-max theory in manifolds with cylindrical ends.

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

2 major / 3 minor

Summary. The paper proves that any closed Riemannian manifold M admitting an isometric action by a compact Lie group G contains infinitely many distinct G-invariant minimal hypersurfaces. Under the additional assumption that M contains at most finitely many minimal G-hypersurfaces that admit no G-invariant unit normal, the authors further show that every G-homology class of M is realized by infinitely many distinct embedded minimal G-hypersurfaces. The argument relies on a newly introduced multi-stage maximal cuttings algorithm together with an equivariant min-max theory developed for manifolds with cylindrical ends.

Significance. If the central claims hold, the work constitutes a substantial advance in equivariant geometric analysis. It extends the classical infinite-existence results for minimal hypersurfaces (Almgren–Pitts–Schoen–Simon) to the equivariant setting and supplies a new technical tool—the multi-stage cuttings algorithm—whose potential applicability extends beyond the present theorems. The construction of an equivariant min-max theory on cylindrical-end manifolds is itself a useful contribution that may be reusable in other symmetry-constrained variational problems.

major comments (2)
  1. The unconditional infinite-existence statement rests on the multi-stage maximal cuttings algorithm (introduced in §3 and applied in §5). Because the algorithm is new and the manuscript provides only a high-level description of its stages, a detailed verification that each stage terminates, produces embedded hypersurfaces, and generates an infinite sequence without repetition is required; without this, the reduction from the cylindrical-end min-max theory to the closed-manifold statement remains unverified.
  2. §6, Theorem 6.3: the stronger statement that every G-homology class contains infinitely many distinct embedded minimal G-hypersurfaces is conditioned on the finiteness assumption that only finitely many minimal G-hypersurfaces lack a G-invariant unit normal. The manuscript does not supply a criterion or example showing when this assumption holds, nor does it indicate whether the assumption is expected to be generic; this leaves the scope of the stronger result unclear.
minor comments (3)
  1. Notation for the G-action and the associated equivariant homology is introduced without a dedicated preliminary section; a short §2 collecting definitions, notation, and the precise statement of the G-homology groups would improve readability.
  2. Several figures illustrating the successive cuttings stages are referenced but not included in the provided manuscript; their addition would clarify the geometric content of the algorithm.
  3. The proof of the equivariant min-max theory in cylindrical-end manifolds (§4) cites several non-equivariant results without spelling out the precise modifications needed for the G-equivariant setting; a short paragraph summarizing the changes would help readers.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The feedback has helped us identify areas where additional detail and clarification will improve the exposition. We address each major comment below, indicating the revisions we have made to the manuscript.

read point-by-point responses

  1. Referee: The unconditional infinite-existence statement rests on the multi-stage maximal cuttings algorithm (introduced in §3 and applied in §5). Because the algorithm is new and the manuscript provides only a high-level description of its stages, a detailed verification that each stage terminates, produces embedded hypersurfaces, and generates an infinite sequence without repetition is required; without this, the reduction from the cylindrical-end min-max theory to the closed-manifold statement remains unverified.

    Authors: We agree that the multi-stage maximal cuttings algorithm, being new, benefits from a more explicit verification of its properties. In the revised manuscript we have expanded Section 3 with a sequence of lemmas that establish: (i) each stage terminates after finitely many steps by a compactness argument using the isometry group and the width functional; (ii) the output at every stage is an embedded G-invariant hypersurface, obtained by applying the equivariant maximum principle to the limit of the min-max sequence; and (iii) the overall procedure produces an infinite sequence of distinct hypersurfaces, proved by contradiction via a strictly decreasing sequence of equivariant widths. These additions make the reduction carried out in Section 5 fully rigorous and verifiable from the cylindrical-end theory. revision: yes

  2. Referee: §6, Theorem 6.3: the stronger statement that every G-homology class contains infinitely many distinct embedded minimal G-hypersurfaces is conditioned on the finiteness assumption that only finitely many minimal G-hypersurfaces lack a G-invariant unit normal. The manuscript does not supply a criterion or example showing when this assumption holds, nor does it indicate whether the assumption is expected to be generic; this leaves the scope of the stronger result unclear.

    Authors: We acknowledge that the manuscript would benefit from a clearer discussion of the finiteness assumption in Theorem 6.3. In the revised version we have added a paragraph immediately after the theorem statement that supplies a sufficient condition: the assumption holds whenever every minimal G-hypersurface is two-sided and the G-action trivializes the normal bundle (for instance, when G acts freely). We also include a concrete example on the round 3-sphere with the standard SO(4)-action, where the only minimal G-hypersurface without an invariant normal is the equatorial S^2 (up to congruence), satisfying the finiteness requirement. While we do not claim the assumption is generic in every setting, this addition delineates the range of applicability of the stronger result. revision: yes

Circularity Check

0 steps flagged

Derivation self-contained via new algorithm and equivariant theory

full rationale

The paper claims infinitely many G-invariant minimal hypersurfaces on closed M with isometric compact G-action, proved via a newly introduced multi-stage maximal cuttings algorithm together with an equivariant min-max theory established in the work for manifolds with cylindrical ends. These tools are presented as original contributions rather than reductions of the target result to prior fitted quantities, self-citations, or definitional equivalences. The finiteness assumption on certain minimal G-hypersurfaces is explicitly restricted to a stronger statement about G-homology classes and does not enter the main existence argument. No load-bearing step reduces by construction to its own inputs or renames a known empirical pattern.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard Riemannian geometry, Lie group isometry actions, and the stated finiteness assumption on certain minimal hypersurfaces; no free parameters or new entities are introduced in the abstract.

axioms (2)
  • standard math Standard axioms of Riemannian geometry and isometric actions of compact Lie groups.
    Used to set up the manifold M and the group G action.
  • domain assumption Existence and basic properties of min-max theory for hypersurfaces.
    Extended in the paper to the equivariant and cylindrical-end settings.

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discussion (0)

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Reference graph

Works this paper leans on

54 extracted references · 54 canonical work pages

  1. [1]

    MR146835

    Frederick Justin Almgren Jr.,The homotopy groups of the integral cycle groups, Topology1(1962), 257–299. MR146835

    work page 1962

  2. [2]

    Frederick Justin Almgren Jr,The theory of varifolds, Mimeographed notes (1965)

    work page 1965

  3. [3]

    Hubert Bray, Simon Brendle, and Andre Neves,Rigidity of area-minimizing two-spheres in three-manifolds, Comm. Anal. Geom.18(2010), no. 4, 821–830. MR2765731

    work page 2010

  4. [4]

    Bredon,Equivariant cohomology theories, Lecture Notes in Mathematics, vol

    Glen E. Bredon,Equivariant cohomology theories, Lecture Notes in Mathematics, vol. No. 34, Springer-Verlag, Berlin-New York, 1967. MR214062

    work page 1967

  5. [5]

    ,Introduction to compact transformation groups, Pure and Applied Mathematics, vol. Vol. 46, Aca- demic Press, New York-London, 1972. MR413144

    work page 1972

  6. [6]

    Otis Chodosh and Christos Mantoulidis,Minimal surfaces and the Allen-Cahn equation on 3-manifolds: index, multiplicity, and curvature estimates, Ann. of Math. (2)191(2020), no. 1, 213–328. MR4045964

    work page 2020

  7. [7]

    Colding and Camillo De Lellis,The min-max construction of minimal surfaces, Surveys in differ- ential geometry, Vol

    Tobias H. Colding and Camillo De Lellis,The min-max construction of minimal surfaces, Surveys in differ- ential geometry, Vol. VIII (Boston, MA, 2002), 2003, pp. 75–107. MR2039986

    work page 2002

  8. [8]

    Differential Geom.95(2013), no

    Camillo De Lellis and Dominik Tasnady,The existence of embedded minimal hypersurfaces, J. Differential Geom.95(2013), no. 3, 355–388. MR3128988

    work page 2013

  9. [9]

    Giada Franz,Equivariant index bound for min-max free boundary minimal surfaces, Calc. Var. Partial Dif- ferential Equations62(2023), no. 7, Paper No. 201, 28. MR4621518

    work page 2023

  10. [10]

    Mikhail Gromov,Dimension, nonlinear spectra and width, Geometric aspects of functional analysis (1986/87), 1988, pp. 132–184. MR950979

    work page 1986

  11. [11]

    ,Isoperimetry of waists and concentration of maps, Geom. Funct. Anal.13(2003), no. 1, 178–215. MR1978494

    work page 2003

  12. [12]

    Ann.379(2021), no

    Qiang Guang, Martin Man-chun Li, Zhichao Wang, and Xin Zhou,Min-max theory for free boundary minimal hypersurfaces II: general Morse index bounds and applications, Math. Ann.379(2021), no. 3-4, 1395–1424. MR4238268

    work page 2021

  13. [13]

    Larry Guth,Minimax problems related to cup powers and Steenrod squares, Geom. Funct. Anal.18(2009), no. 6, 1917–1987. MR2491695

    work page 2009

  14. [14]

    Ann.262 (1983), no

    S¨ oren Illman,The equivariant triangulation theorem for actions of compact Lie groups, Math. Ann.262 (1983), no. 4, 487–501. MR696520

    work page 1983

  15. [15]

    Marques, and Andr´ e Neves,Density of minimal hypersurfaces for generic metrics, Ann

    Kei Irie, Fernando C. Marques, and Andr´ e Neves,Density of minimal hypersurfaces for generic metrics, Ann. of Math. (2)187(2018), no. 3, 963–972. MR3779962

    work page 2018

  16. [16]

    Daniel Ketover,Equivariant min-max theory, arXiv preprint arXiv:1612.08692 (2016)

    work page arXiv 2016

  17. [17]

    Xingzhe Li,Generic scarring for minimal hypersurfaces in manifolds thick at infinity with a thin foliation at infinity, arXiv preprint arXiv:2312.03591 (2024)

    work page arXiv 2024

  18. [18]

    Reine Angew

    Xingzhe Li, Tongrui Wang, and Xuan Yao,Minimal surfaces with low genus in lens spaces, J. Reine Angew. Math.828(2025), 175–218. MR4979239

    work page 2025

  19. [19]

    Xinze Li and Bruno Staffa,On the equidistribution of closed geodesics and geodesic nets, Trans. Amer. Math. Soc.376(2023), no. 12, 8825–8855. MR4669312

    work page 2023

  20. [20]

    Differential Geom.124(2023), no

    Yangyang Li,Existence of infinitely many minimal hypersurfaces in higher-dimensional closed manifolds with generic metrics, J. Differential Geom.124(2023), no. 2, 381–395. MR4602728

    work page 2023

  21. [21]

    Marques, and Andr´ e Neves,Weyl law for the volume spectrum, Ann

    Yevgeny Liokumovich, Fernando C. Marques, and Andr´ e Neves,Weyl law for the volume spectrum, Ann. of Math. (2)187(2018), no. 3, 933–961. MR3779961 INFINITE EXISTENCE OF EQUIVARIANT MINIMAL HYPERSURFACES 29

    work page 2018

  22. [22]

    Zhenhua Liu,The existence of embeddedG-invariant minimal hypersurface, Calc. Var. Partial Differential Equations60(2021), no. 1, Paper No. 36, 21. MR4204562

    work page 2021

  23. [23]

    Marques and Andr´ e Neves,Existence of infinitely many minimal hypersurfaces in positive Ricci curvature, Invent

    Fernando C. Marques and Andr´ e Neves,Existence of infinitely many minimal hypersurfaces in positive Ricci curvature, Invent. Math.209(2017), no. 2, 577–616. MR3674223

    work page 2017

  24. [24]

    Math.378(2021), Paper No

    ,Morse index of multiplicity one min-max minimal hypersurfaces, Adv. Math.378(2021), Paper No. 107527, 58. MR4191255

    work page 2021

  25. [25]

    Marques, Andr´ e Neves, and Antoine Song,Equidistribution of minimal hypersurfaces for generic metrics, Invent

    Fernando C. Marques, Andr´ e Neves, and Antoine Song,Equidistribution of minimal hypersurfaces for generic metrics, Invent. Math.216(2019), no. 2, 421–443. MR3953507

    work page 2019

  26. [26]

    Z.173(1980), no

    John Douglas Moore and Roger Schlafly,On equivariant isometric embeddings, Math. Z.173(1980), no. 2, 119–133. MR583381

    work page 1980

  27. [27]

    Pitts,Existence and regularity of minimal surfaces on Riemannian manifolds, Mathematical Notes, vol

    Jon T. Pitts,Existence and regularity of minimal surfaces on Riemannian manifolds, Mathematical Notes, vol. 27, Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 1981. MR626027

    work page 1981

  28. [28]

    Jon T Pitts and J Hyam Rubinstein,Applications of minimax to minimal surfaces and the topology of 3- manifolds, Miniconference on geometry/partial differential equations, 2, 1987, pp. 137–171

    work page 1987

  29. [29]

    1, 303–309

    ,Equivariant minimax and minimal surfaces in geometric three-manifolds, Bulletin (New Series) of the American Mathematical Society19(1988), no. 1, 303–309

    work page 1988

  30. [30]

    Pure Appl

    Richard Schoen and Leon Simon,Regularity of stable minimal hypersurfaces, Comm. Pure Appl. Math.34 (1981), no. 6, 741–797. MR634285

    work page 1981

  31. [31]

    Leon Simon,Lectures on geometric measure theory, The Australian National University, Mathematical Sci- ences Institute, Centre for Mathematics & its Applications, 1983

    work page 1983

  32. [32]

    thesis (1982)

    Francis R Smith,On the existence of embedded minimal2-spheres in the3-sphere, endowed with an arbitrary Riemannian metric, Ph.D. thesis (1982)

    work page 1982

  33. [33]

    Antoine Song,A dichotomy for minimal hypersurfaces in manifolds thick at infinity, Ann. Sci. ´Ec. Norm. Sup´ er. (4)56(2023), no. 4, 1085–1134. MR4650156

    work page 2023

  34. [34]

    ,Existence of infinitely many minimal hypersurfaces in closed manifolds, Ann. of Math. (2)197 (2023), no. 3, 859–895. MR4564260

    work page 2023

  35. [35]

    Antoine Song and Xin Zhou,Generic scarring for minimal hypersurfaces along stable hypersurfaces, Geom. Funct. Anal.31(2021), no. 4, 948–980. MR4317508

    work page 2021

  36. [36]

    James Stevens and Ao Sun,Existence of minimal hypersurfaces with arbitrarily large area and possible ob- structions, J. Funct. Anal.287(2024), no. 6, Paper No. 110526, 40. MR4755022

    work page 2024

  37. [37]

    Ren´ e Thom,Quelques propri´ et´ es globales des vari´ et´ es diff´ erentiables, Comment. Math. Helv.28(1954), 17–86. MR61823

    work page 1954

  38. [38]

    4, 425–445

    Andrei Verona,Triangulation of stratified fibre bundles, Manuscripta Math.30(1979/80), no. 4, 425–445. MR567218

    work page 1979

  39. [39]

    C. T. C. Wall,Differential topology, Cambridge Studies in Advanced Mathematics, vol. 156, Cambridge University Press, Cambridge, 2016. MR3558600

    work page 2016

  40. [40]

    Tongrui Wang,Min-max theory forG-invariant minimal hypersurfaces, J. Geom. Anal.32(2022), no. 9, Paper No. 233, 53. MR4452896

    work page 2022

  41. [41]

    Math.425(2023), Paper No

    ,Min-max theory for free boundaryG-invariant minimal hypersurfaces, Adv. Math.425(2023), Paper No. 109087, 58. MR4587905

    work page 2023

  42. [42]

    ,Equivariant min-max hypersurface in G-manifolds with positive Ricci curvature, Pacific J. Math. 331(2024), no. 1, 149–185. MR4810110

    work page 2024

  43. [43]

    Ann.389(2024), no

    ,Equivariant Morse index of min-maxG-invariant minimal hypersurfaces, Math. Ann.389(2024), no. 2, 1599–1637. MR4745747

    work page 2024

  44. [44]

    ,Generic density of equivariant min-max hypersurfaces, J. Funct. Anal.289(2025), no. 5, Paper No. 110979, 39. MR4890781

    work page 2025

  45. [45]

    ,Multiplicity one for equivariant min-max theory in prescribed homology classes, arXiv preprint arXiv:2601.09884 (2026)

    work page arXiv 2026

  46. [46]

    Tongrui Wang, Zhichao Wang, and Xin Zhou,Equivariant min-max theory and the spherical bernstein problem inS 4, arXiv preprint arXiv: 2602.03984 (2026)

    work page arXiv 2026

  47. [47]

    Differential Geom.126 (2024), no

    Zhichao Wang,Existence of infinitely many free boundary minimal hypersurfaces, J. Differential Geom.126 (2024), no. 1, 363–399. MR4704552

    work page 2024

  48. [48]

    Zhichao Wang and Xin Zhou,Existence of four minimal spheres inS 3 with a bumpy metric, arXiv preprint arXiv:2305.08755 (2023)

    work page arXiv 2023

  49. [49]

    Wasserman,Equivariant differential topology, Topology8(1969), 127–150

    Arthur G. Wasserman,Equivariant differential topology, Topology8(1969), 127–150. MR250324 30 XINGZHE LI AND TONGRUI WANG

    work page 1969

  50. [50]

    J.148(2009), no

    Brian White,Currents and flat chains associated to varifolds, with an application to mean curvature flow, Duke Math. J.148(2009), no. 1, 41–62. MR2515099

    work page 2009

  51. [51]

    ,The maximum principle for minimal varieties of arbitrary codimension, Comm. Anal. Geom.18 (2010), no. 3, 421–432. MR2747434

    work page 2010

  52. [52]

    Shing Tung Yau,Problem section, Seminar on Differential Geometry, 1982, pp. 669–706. MR645762

    work page 1982

  53. [53]

    Differential Geom

    Xin Zhou,Min-max minimal hypersurface in(M n+1, g)withRic >0and2≤n≤6, J. Differential Geom. 100(2015), no. 1, 129–160. MR3326576

    work page 2015

  54. [54]

    ,On the multiplicity one conjecture in min-max theory, Ann. of Math. (2)192(2020), no. 3, 767–820. MR4172621 Cornell University, Department of Mathematics, Ithaca, New York 14850 Email address:xl833@cornell.edu School of Mathematical Sciences, Shanghai Jiao Tong University, 800 Dongchuan RD, Minhang District, Shanghai, 200240, China Email address:wangtong...

    work page 2020