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Area-minimizing submanifolds are not generically smooth, except for geodesics, minimal surfaces, and minimal hypersurfaces

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Pith's one-line read Area-minimizing mod-2 submanifolds are not generically smooth once dimension and codimension both exceed the classical cases.

desk verdict Settles White’s mod-2 generic-smoothness question outside the three classical regimes by proving the Veronese RP^{2} cone is mod-2 minimizing and realizing persistent (d-3)-singular sets via a new gluing construction. read the letter →

arxiv 2607.24735 v2 pith:WZMOY3P5 submitted 2026-07-27 math.DG

Zhenhua Liu This is my paper
classification math.DG MSC 49Q1553C4249Q0558A35
keywords area-minimizingcurrentsmod2homologygenericregularityVeroneseconepersistentsingularitiesmoderationsWhiteconjecture
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

White asked whether singularities of homologically area-minimizing cycles disappear after a generic change of the ambient metric. For mod-2 homology the answer is no outside three classical regimes: geodesics, surfaces, and hypersurfaces. The paper constructs, for every nonzero d-dimensional class with d≥3 and codimension at least 2, a nonempty open set of smooth metrics in which every area-minimizing representative is forced to keep a singular set of Hausdorff dimension at least d−3. The engine is a new proof that the cone on the Veronese embedding of RP² is itself mod-2 area-minimizing, together with a metric-gluing procedure that inserts products of that cone with spheres into any given homology class while keeping the inserted singularities stable under small metric perturbations. The result is sharp relative to the known positive theorems in dimensions 1–2 and codimension 1, and it shows that topological non-bounding of RP² already produces persistent geometric singularities three dimensions earlier than Thom’s classical topological obstruction.

What carries the argument

Moderations: a projection-integral criterion (Definition 3.0.1) that certifies a compactly supported mod-2 current is area-minimising. The truncated Veronese cone is shown to be a moderation by combining coherent-state inequalities from representation theory with an explicit computation of its projected volume; products of that cone with spheres then supply the persistent singular sets that are glued into arbitrary classes.

What would settle it

Exhibit a mod-2 current of strictly smaller mass than the truncated Veronese cone that has the same boundary (the Veronese RP² itself), or show that the representation-theoretic integral bound used to prove the moderation inequality is false for some three-plane.

Watch

Extended reading notes

Core claim

For any nonzero d-dimensional mod-2 homology class on a compact (d+c)-manifold with d≥3 and c≥2 there exists a nonempty open set of smooth Riemannian metrics such that every area-minimizing representative has singular set of Hausdorff dimension at least 0, and at least d−3 whenever the class admits a smoothly embedded representative. The same construction realises, as unique minimisers, currents whose singular sets are standard spheres and whose tangent cones are regular cylindrical products of the Veronese cone.

Load-bearing premise

The entire persistence argument rests on the claim that the cone over the Veronese minimal embedding of RP² is mod-2 area-minimising; if that cone fails to minimise, the forced singular sets of dimension d−3 disappear.

Editorial extensions

If this is right

  • Outside dimensions 1–2 and codimension 1, generic metrics cannot eliminate singularities of mod-2 area-minimisers.
  • Any homology class that admits a smooth representative still forces singular sets of dimension at least d−3 on an open set of metrics.
  • The same gluing realises prescribed regular cylindrical tangent cones (including products of the Veronese cone with spheres) as the unique minimisers of their classes.
  • Analogous statements hold for Z/3Z coefficients once the triple junction is used in place of the Veronese cone, with singular-set dimension at least d−1.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The moderation criterion may settle other open area-minimising cones (for instance remaining isoparametric focal cones) without calibrations or Lawlor retractions.
  • Because the obstruction is already present at d=3, one expects that stationary (not merely minimising) mod-2 varifolds likewise fail to be generically smooth in the same range.
  • The metric-shorting-plus-squeezing gluing technique offers a calibration-free route to realisation theorems for singular minimisers in other coefficient groups.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 10 minor

Summary. The paper proves that mod-2 area-minimizing submanifolds are not generically smooth outside the known exceptional cases (d=1 geodesics, d=2 surfaces, c=1 hypersurfaces), settling a conjecture of White. Theorem 1 asserts that for every nonzero d-dimensional mod-2 class on a compact (d+c)-manifold with d≥3, c≥2, there is a nonempty open set of smooth metrics in which every area-minimizing representative is singular, with singular set of Hausdorff dimension at least d−3 when the class has a smooth embedded representative. The decisive new input is Theorem 2: the cone over the Veronese minimal embedding of RP² in S⁴ is mod-2 area-minimizing, resolving a question open since the 1980s. This is proved via a new criterion, "moderation" (Definition 3.0.1, Theorem 4), which reduces minimization to a family of sharp projection-area inequalities verified in §4 using coherent-state inequalities from representation theory (Lieb–Solovej, Frank, Kulikov et al.) and an explicit computation of the projection of the truncated cone onto its tangent plane (Lemma 4.5.3, an ellipsoid of volume 3π/4). Theorems 1 and 3 then follow by topologically inserting σ_s(C) into a representative (connected sums, regular neighborhoods, §§5–7), a metric-gluing argument that makes the altered current uniquely minimizing (§8), and a persistence/slicing argument (§9) exploiting that RP² does not bound a smooth 3-manifold.

Significance. If correct — and the load-bearing computation appears to be — this is a landmark result: it gives a sharp negative answer to White's conjecture, identifies the exact dimension threshold (d−3) for persistent singularities, and solves a problem open since the 1980s about the Veronese cone. Two strengths deserve explicit mention. First, the key step is not an abstract compactness argument but an explicit, parameter-free computation: the moderated mass is pinned down to 3π/4 by a provably tight inequality chain, and the author ships Mathematica verification of the algebraic identities (e.g., the orthonormal basis of V), which materially de-risks the normalization-heavy parts. Second, the "moderation" criterion (Definition 3.0.1, Theorem 4) is a genuinely new, falsifiable sufficient condition for mod-2 area-minimization that upgrades Morgan's calibration-mod-ν ideas and is likely to have independent applications (the author mentions isoparametric focal cones). The gluing construction in §8, which forces uniqueness without calibrations or Lawlor retractions, and Theorem 3 (realization of tangent cones without calibrations) are further independent contributions. The result is outside the

major comments (3)
  1. [§4.4, Fact 4.4.1 and Eq. (4.5)] Fact 4.4.1 is the load-bearing external input for the entire paper: it supplies the upper bound in the zero-slack squeeze (4.31) on which Theorem 2, and hence Theorems 1 and 3, rest. Two points need to be made explicit. (i) The step from (4.4) to (4.5) is dismissed as 'just a change of variables,' but it produces the density πs^{−3/4}, and every subsequent constant (the split at 1/16 in (4.16)–(4.18), the value 3/4 in (4.21)) depends on it. Please write out the computation: the pushforward of spherical measure under the spin-2 coherent-state overlap s=((1+z)/2)^4, identifying it with the J=2 case of [32, Thm 4]. (ii) [32, Thm 4] is invoked for the convex but non-smooth function Φ(s)=|s−1/16|; please state the precise hypotheses of [32, Thm 4] and either confirm it covers all convex Φ or insert a mollification/limiting argument. Neither point is likely to fail — the split point 1/16 is th
  2. [§9.3, proof of Lemma 9.2.2] This is where the (d−3) lower bound is actually won, and it is the most compressed argument in the paper. Please expand the justification of two claims: (a) that for H^{d−3}-a.e. p the slice T_p = ⟨T, π_{S^{d−3}}, p⟩ has boundary exactly RP² — this requires the boundary-of-slice formula for the current T restricted to U_{2ϵ}(Sing σ_{d−3}(C)), together with the constancy theorem applied in the annular region where Lemma 9.1.1 gives graphicality, and the assertion that the relevant slice of the section S_T is diffeomorphic to RP²; (b) the deduction from 'π^{−1}(p)∩Supp T lies in Reg T' to 'T_q is a smooth compact 3-manifold with boundary RP² for a.e. q near p' via Sard's theorem. As written, the contradiction with the non-bounding of RP² ([26, Lemma 7.1]) is correct in outline but the slicing regularity hypotheses ([27, 4.3.6, 4.3.13]) are invoked at a level of generality that deserves at
  3. [§8.5.4, Fact 8.5.8 and its use of [52, Lemma 2.4.1]] The uniqueness conclusion T = N#σ_s(C) is obtained by applying [52, Lemma 2.4.1], which is stated in the integral-current setting, to deduce that two mod-2 currents agreeing on a relatively open set agree everywhere via infinite-order tangency. Elsewhere in the manuscript (e.g., Lemmas 2.7.2, 6.1.3, 9.1.1) the author helpfully reproduces the [52] arguments adapted to mod 2; here the adaptation is only cited. Please either state the mod-2 version of [52, Lemma 2.4.1] with a proof sketch, or explain why the integral-current statement transports directly. The issue is load-bearing because uniqueness in h_β is what upgrades Fact 8.5.10 into Lemma 8.0.1.
minor comments (10)
  1. [§1, passim] Numbering: the text repeatedly refers to 'Theorems 1 and 1' (e.g., p. 3 twice, §1.1 'Theorem 1 is a direct corollary of Theorem 1'). Presumably one of these is 'Result 1' or Theorem 3; please repair the cross-references.
  2. [Theorem 3 statement] The statement says the current 'equals C truncated times S^2' near its singular set; per Definition 1.0.5 and Lemma 8.0.1 the factor should be S^s.
  3. [Lemma 4.5.3, statement] The ellipsoid in the displayed statement is written with first term u²/(3/4) and second v²/(3/4), but (4.28) shows the first two terms should be v²/(3/4) and w²/(3/4) in the (u,v,w) coordinates.
  4. [Proof of Lemma 4.5.3] In the embedding argument, the conclusion 'x′ = ±y' should read 'x′ = ±x'.
  5. [§4.6, first sentence] The sentence 'we deduce that , we need to determine for any point p ∈ C1, we have M(π_{TpC}(C1)) = 3π/4' is garbled.
  6. [Acknowledgements] The acknowledgements contain a scrambled sentence ('The author would like to thank Last but not least, the author would also like to Professors Yongsheng Zhang and Zhihan Wang ... express immense gratitude for Professor Hubert Bray's ...').
  7. [Passim] Typos: 'Techincally' (§2.3), 'interchangbly' (§1), 'In fact its is' (§1), 'warp up' (§1.2), 'stablizers' (Fact 4.1.5), 'Lie algbera so(3)' (§4.3 heading), 'conlusion' in the citation of [39], duplicated 'coarea formula coarea formula' (Fact 8.3.1), stray ']' after (8.18), duplicated 'thatCis thatCis' (after Definition 4.1.6).
  8. [Figure 3 caption, §8.2] The editorial aside 'We emphasize that Reg(N#σ_s(C)) should be connected! Impossible to draw on a 2-d plane.' should be removed or rephrased for the published version.
  9. [References] Reference [70] (White) appears with empty journal field 'In: ()'; please complete the bibliographic data.
  10. [§8.3, Eq. (8.4)] In Fact 8.3.1, M(C1 × S^1) should be M(C1 × S^s) in (8.4).

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: self-contained existence proof; moderation criterion and Veronese verification are independent calculations against external inequalities, not definitional or fitted loops.

full rationale

This is a pure GMT existence/proof paper. Theorem 4 (moderation implies mod-2 minimizing) is derived from first principles via orthogonal projections, Fact 2.4.2/2.4.3, and Fubini (eqs. 3.3–3.9). Theorem 2 applies it to the Veronese cone C1 by verifying Def. 3.0.1 via (i) explicit SO(3)/F-parametrization and orthonormal frames (Facts 4.1.x–4.5.x, Mathematica check of κ-orthonormality), (ii) external coherent-state inequalities of Lieb–Solovej/Frank/Kulikov (Fact 4.4.1 citing [32,50,45], with Lie-algebra conventions translated) giving the integral upper bound ≤3π/4 (Lemma 4.4.2), and (iii) direct computation that the projected mass equals exactly 3π/4 (Lemma 4.5.3 ellipsoid). The squeeze to equality (4.31) is a verification that the conditions hold, not a fit or self-definition. Subsequent constructions (σs(C), metric gluing via squeezing map f and shorting Lemma 8.1.1, Allard persistence) take the minimizing cone as an established input and build open sets of metrics; they do not feed back into the cone verification. Self-citations to the author’s integral paper [52] supply parallel topological/gluing lemmas whose proofs are fully reproduced in the text (e.g., Lemmas 2.7.2, 6.1.3, 7.0.1, 9.1.1) and are coefficient-independent; they are not load-bearing uniqueness black boxes. No parameter is fitted to data and re-predicted; no ansatz is smuggled; no result is renamed. The derivation chain is therefore non-circular.

Assumptions & free parameters 0 free parameters · 6 assumptions · 2 invented entities

The argument rests on standard GMT existence/regularity, classical algebraic topology (RP2 does not bound), and one external sharp inequality from representation theory. No free parameters are fitted. The only invented notions are the moderation criterion and the regular-cylindrical cone class, both given explicit definitions and used instrumentally.

assumptions (6)
  • standard math Federer–Fleming: every mod-2 homology class on a compact Riemannian manifold admits an area-minimizing representative.
    Invoked from the first paragraph; classical existence theorem.
  • standard math Simon / Federer regularity: mod-2 area-minimizers are smooth outside a codimension-2 countably rectifiable singular set.
    Used to justify calling minimizers ‘submanifolds’ and to bound spine dimension s≤d−2.
  • standard math Coherent-state / Lieb–Solovej-type inequality for the irreducible 5-dimensional representation of SO(3)/SU(2) (Fact 4.4.1).
    External analytic input that supplies the sharp integral bound needed to verify the moderation inequalities for the Veronese cone.
  • standard math RP2 does not bound any compact 3-manifold (mod-2 Euler characteristic obstruction).
    Topological fact used in §9 to force interior singularities in almost every slice.
  • standard math Whitney stratified sets are triangulable; simplicial and smooth regular-neighborhood theorems (Goresky, Hudson, Hirsch).
    Used in §7 to produce arbitrarily small regular neighborhoods of the singular representative.
  • standard math Allard’s regularity theorem upgrades mass-close stationary varifolds to C^4 graphical perturbations.
    Lemma 9.1.1; converts flat/varifold closeness into the graphical control needed for slicing.
invented entities (2)
  • Moderation (Definition 3.0.1)
    purpose: Integral-projection criterion that implies mod-2 area-minimization without a calibration form.
    New sufficient condition; verified for the Veronese cone by direct computation. No independent experimental handle required (pure math).
  • Regular cylindrical mod-2 area-minimizing cone (Definition 1.0.5)
    purpose: Class of cones whose singular spines can be realized as persistent singular sets via the gluing theorem.
    Organizational definition for the realization theorem (Thm 3); not an ontological postulate.

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Pith. "Pith review of Area-minimizing submanifolds are not generically smooth, except for geodesics, minimal surfaces, and minimal hypersurfaces." pith.science (2026). https://pith.science/paper/WZMOY3P5

@misc{pith2026260724735,
  author       = {Pith},
  title        = {Pith review of: Area-minimizing submanifolds are not generically smooth, except for geodesics, minimal surfaces, and minimal hypersurfaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WZMOY3P5}},
  note         = {Machine review of arXiv:2607.24735}
}
abstract

We prove that area-minimizing submanifolds in mod $2$ homology are not generically smooth, except in the case of geodesics, minimal surfaces and minimal hypersurfaces. This settles a conjecture of White that asks the generic smoothness of area-minimizing submanifolds in mod $2$ homology. We furthermore establish a lower bound on the Hausdorff dimension of the singular sets of area-minimizing submanifolds with respect to open sets of Riemannian metrics. The lower bound is ${(d-3)},$ where $d$ denotes the dimension of the submanifold. As a crucial step, we prove that the cone over the Veronese minimal embedding of $\rpt$ is mod $2$ area-minimizing, settling another long-standing open problem.

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Works this paper leans on

76 extracted references · 12 linked inside Pith

  1. [52]

    Area-minimizing submanifolds are not generically smooth

    Zhenhua Liu. “Area-minimizing submanifolds are not generically smooth”. In:Duke Math. J.175.6 (2026), pp. 959–1066

    2026

  2. [1]

    On the first variation of a varifold

    William K. Allard. “On the first variation of a varifold”. In:Ann. of Math. (2)95 (1972), pp. 417–491

    1972

  3. [2]

    Almgren Jr.Almgren’s big regularity paper

    Frederick J. Almgren Jr.Almgren’s big regularity paper. Vol. 1. World Scientific Monograph Series in Mathematics.Q-valued functions minimizing Dirichlet’s integral and the regularity of area-minimizing rectifiable currents up to codimension 2, With a preface by Jean E. Taylor and Vladimir Scheffer. World Scientific Publishing Co. Inc. River Edge NJ, 2000,...

    2000

  4. [3]

    Fredrick Almgren et al.Optimal smooth approximation of integral cycles. 2024. arXiv:2411 . 17678 [math.DG]. 46 REFERENCES

    2024

  5. [4]

    Andreas Arvanitoyeorgos.An introduction to Lie groups and the geometry of homogeneous spaces. Vol. 22. Student Mathematical Library. Translated from the 1999 Greek original and revised by the author. American Mathematical Society, Providence, RI, 2003, pp. xvi+141

    1999

  6. [5]

    Yuchen Bi et al.A proof for the Riemannian positive mass theorem up to dimension 19. 2026. arXiv: 2603.02769 [math.DG]

    arXiv 2026

  7. [6]

    Brendle and Y

    S. Brendle and Y. Wang.A dimension descent scheme for the positive mass theorem in arbitrary dimension. 2026. arXiv:2604.08473 [math.DG]

    arXiv 2026

  8. [7]

    Theodor Br ¨ocker and Tammo tom Dieck.Representations of compact Lie groups. Vol. 98. Grad- uate Texts in Mathematics. Springer-Verlag, New York, 1985, pp. x+313

    1985

Show all 76 references
  1. [8]

    Some open problems in geometric measure theory and its applications suggested by par- ticipants of the 1984 AMS summer institute

    “Some open problems in geometric measure theory and its applications suggested by par- ticipants of the 1984 AMS summer institute”. In:Geometric measure theory and the calculus of variations (Arcata, Calif., 1984). Ed. by J. E. Brothers. Vol. 44. Proc. Sympos. Pure Math. Amer....

    1984

  2. [9]

    Gianmarco Caldini.On smooth approximation of integral cycles mod 2. 2025. arXiv:2511.10545 [math.DG]

    2025

  3. [10]

    Alessandro Carlotto, Yangyang Li, and Zhihan Wang.Non-persistence of strongly isolated sin- gularities, and geometric applications. 2025. arXiv:2411.12677 [math.DG]

    2025 arXiv

  4. [11]

    Two-dimensional area minimizing integral currents are classical minimal surfaces

    Sheldon Xu-Dong Chang. “Two-dimensional area minimizing integral currents are classical minimal surfaces”. In:J. Amer. Math. Soc.1.4 (1988), pp. 699–778

    1988

  5. [12]

    Minimal submanifolds of a sphere with second fundamental form of constant length

    S. S. Chern, M. do Carmo, and S. Kobayashi. “Minimal submanifolds of a sphere with second fundamental form of constant length”. In:Functional Analysis and Related Fields (Proc. Conf. for M. Stone, Univ. Chicago, Chicago, Ill., 1968). Springer, New York-Berlin, 1970, pp. 59–75

    1968

  6. [13]

    Generic regularity for minimizing hypersurfaces in dimensions 9 and 10

    Otis Chodosh, Christos Mantoulidis, and Felix Schulze. “Generic regularity for minimizing hypersurfaces in dimensions 9 and 10”. In:Publ. Math. Inst. Hautes ´Etudes Sci.143 (2026), pp. 143–188

    2026

  7. [14]

    Improved generic regularity of codimension-1 minimizing integral currents

    Otis Chodosh, Christos Mantoulidis, and Felix Schulze. “Improved generic regularity of codimension-1 minimizing integral currents”. In:Ars Inven. Anal.(2024), Paper No. 3, 16

    2024

  8. [15]

    Otis Chodosh et al.Generic regularity for minimizing hypersurfaces in dimension 11. 2025. arXiv: 2506.12852 [math.DG]

    2025 arXiv

  9. [16]

    Uniqueness of Tangent Cones

    Camillo De Lellis, Paul Minter, and Anna Skorobogatova.The Fine Structure of the Singu- lar Set of Area-Minimizing Integral Currents III: Frequency 1 Flat Singular Points andH m−2-a.e. Uniqueness of Tangent Cones. 2024. arXiv:2304.11553 [math.AP]

    2024 arXiv

  10. [17]

    Camillo De Lellis and Anna Skorobogatova.The fine structure of the singular set of area-minimizing integral currents I: the singularity degree of flat singular points. 2023. arXiv:2304.11552 [math.AP]

    2023 arXiv

  11. [18]

    Camillo De Lellis and Anna Skorobogatova.The fine structure of the singular set of area-minimizing integral currents II: rectifiability of flat singular points with singularity degree larger than1. 2023. arXiv:2304.11555 [math.AP]

    2023 arXiv

  12. [19]

    Regularity of area minimizing currents I: gradi- entL p estimates

    Camillo De Lellis and Emanuele Spadaro. “Regularity of area minimizing currents I: gradi- entL p estimates”. In:Geom. Funct. Anal.24.6 (2014), pp. 1831–1884

    2014

  13. [20]

    Regularity of area minimizing currents II: center manifold

    Camillo De Lellis and Emanuele Spadaro. “Regularity of area minimizing currents II: center manifold”. In:Ann. of Math. (2)183.2 (2016), pp. 499–575

    2016

  14. [21]

    Regularity of area minimizing currents III: blow- up

    Camillo De Lellis and Emanuele Spadaro. “Regularity of area minimizing currents III: blow- up”. In:Ann. of Math. (2)183.2 (2016), pp. 577–617

    2016

  15. [22]

    Regularity theory for2-dimensional almost minimal currents I: Lipschitz approximation

    Camillo De Lellis, Emanuele Spadaro, and Luca Spolaor. “Regularity theory for2-dimensional almost minimal currents I: Lipschitz approximation”. In:Trans. Amer. Math. Soc.370.3 (2018), pp. 1783–1801. REFERENCES 47

    2018

  16. [23]

    Regularity theory for 2-dimensional almost minimal currents II: Branched center manifold

    Camillo De Lellis, Emanuele Spadaro, and Luca Spolaor. “Regularity theory for 2-dimensional almost minimal currents II: Branched center manifold”. In:Ann. PDE3.2 (2017), Paper No. 18, 85

    2017

  17. [24]

    Regularity theory for 2-dimensional almost minimal currents III: Blowup

    Camillo De Lellis, Emanuele Spadaro, and Luca Spolaor. “Regularity theory for 2-dimensional almost minimal currents III: Blowup”. In:J. Differential Geom.116.1 (2020), pp. 125–185

    2020

  18. [25]

    J. J. Duistermaat and J. A. C. Kolk.Lie groups. Universitext. Springer-Verlag, Berlin, 2000, pp. viii+344

    2000

  19. [26]

    Euler flag enumeration of Whit- ney stratified spaces

    Richard Ehrenborg, Mark Goresky, and Margaret Readdy. “Euler flag enumeration of Whit- ney stratified spaces”. In:Adv. Math.268 (2015), pp. 85–128

    2015

  20. [27]

    Herbert Federer.Geometric measure theory. Vol. Band 153. Die Grundlehren der mathematis- chen Wissenschaften. Springer-Verlag New York, Inc., New York, 1969, pp. xiv+676

    1969

  21. [28]

    Some theorems on integral currents

    Herbert Federer. “Some theorems on integral currents”. In:Trans. Amer. Math. Soc.117 (1965), pp. 43–67

    1965

  22. [29]

    The singular sets of area minimizing rectifiable currents with codimension one and of area minimizing flat chains modulo two with arbitrary codimension

    Herbert Federer. “The singular sets of area minimizing rectifiable currents with codimension one and of area minimizing flat chains modulo two with arbitrary codimension”. In:Bull. Amer. Math. Soc.76 (1970), pp. 767–771

    1970

  23. [30]

    Normal and integral currents

    Herbert Federer and Wendell H. Fleming. “Normal and integral currents”. In:Ann. of Math. (2)72 (1960), pp. 458–520

    1960

  24. [31]

    Flat chains over a finite coefficient group

    Wendell H. Fleming. “Flat chains over a finite coefficient group”. In:Trans. Amer. Math. Soc. 121 (1966), pp. 160–186

    1966

  25. [32]

    Sharp inequalities for coherent states and their optimizers

    Rupert L. Frank. “Sharp inequalities for coherent states and their optimizers”. In:Adv. Non- linear Stud.23.1 (2023), Paper No. 20220050, 28

    2023

  26. [33]

    Triangulation of stratified objects

    R. Mark Goresky. “Triangulation of stratified objects”. In:Proc. Amer. Math. Soc.72.1 (1978), pp. 193–200

    1978

  27. [34]

    Alfred Gray.Tubes. Second. Vol. 221. Progress in Mathematics. With a preface by Vicente Miquel. Birkh¨auser Verlag, Basel, 2004, pp. xiv+280

    2004

  28. [35]

    Area minimizing hypersurfaces with isolated singularities

    Robert Hardt and Leon Simon. “Area minimizing hypersurfaces with isolated singularities”. In:J. Reine Angew. Math.362 (1985), pp. 102–129

    1985

  29. [36]

    Obstructions to special Lagrangian desingularizations and the Lagrangian prescribed boundary problem

    Mark Haskins and Tommaso Pacini. “Obstructions to special Lagrangian desingularizations and the Lagrangian prescribed boundary problem”. In:Geom. Topol.10 (2006), pp. 1453–1521

    2006

  30. [37]

    Cambridge University Press, Cambridge, 2002, pp

    Allen Hatcher.Algebraic topology. Cambridge University Press, Cambridge, 2002, pp. xii+544

    2002

  31. [38]

    Hirsch.Differential topology

    Morris W. Hirsch.Differential topology. Vol. 33. Graduate Texts in Mathematics. Corrected reprint of the 1976 original. Springer-Verlag, New York, 1994, pp. x+222

    1976

  32. [39]

    Smooth regular neighborhoods

    Morris W. Hirsch. “Smooth regular neighborhoods”. In:Ann. of Math. (2)76 (1962), pp. 524– 530

    1962

  33. [40]

    J. F. P . Hudson.Piecewise linear topology. University of Chicago Lecture Notes prepared with the assistance of J. L. Shaneson and J. Lees. W. A. Benjamin, Inc., New York-Amsterdam, 1969, pp. ix+282

    1969

  34. [41]

    Alexander Kirillov Jr.An introduction to Lie groups and Lie algebras. Vol. 113. Cambridge Stud- ies in Advanced Mathematics. Cambridge University Press, Cambridge, 2008, pp. xii+222

    2008

  35. [42]

    Brian Krummel and Neshan Wickramasekera.Analysis of singularities of area minimizing cur- rents, Part I: planar frequency, branch points of rapid decay, and weak locally uniform approximation

  36. [43]

    Brian Krummel and Neshan Wickramasekera.Analysis of singularities of area-minimizing cur- rents, Part II: a uniform height bound, estimates away from branch points of rapid decay, and unique- ness of tangent cones. 2026. arXiv:2304.10272 [math.DG]. 48 REFERENCES

    2026 arXiv

  37. [44]

    Brian Krummel and Neshan Wickramasekera.Analysis of singularities of area-minimizing cur- rents, Part III: branch points of planar frequency̸=2, higher order asymptotics, and the local topol- ogy. 2026. arXiv:2607.13356 [math.DG]

    2026 arXiv

  38. [45]

    A monotonicity theorem for subharmonic functions on manifolds

    Aleksei Kulikov et al. “A monotonicity theorem for subharmonic functions on manifolds”. In:Adv. Math.479 (2025), Paper No. 110423, 18

    2025

  39. [46]

    A sufficient criterion for a cone to be area-minimizing

    Gary R. Lawlor. “A sufficient criterion for a cone to be area-minimizing”. In:Mem. Amer. Math. Soc.91.446 (1991), pp. vi+111

    1991

  40. [47]

    A note about the Veronese cone

    Gary R. Lawlor and Timothy A. Murdoch. “A note about the Veronese cone”. In:Illinois J. Math.39.2 (1995), pp. 271–277

    1995

  41. [48]

    Lee.Introduction to smooth manifolds

    John M. Lee.Introduction to smooth manifolds. Second. Vol. 218. Graduate Texts in Mathemat- ics. Springer, New York, 2013, pp. xvi+708

    2013

  42. [49]

    Yangyang Li and Zhihan Wang.Minimal hypersurfaces for generic metrics in dimension 8. 2022. arXiv:2205.01047 [math.DG]

    2022 arXiv

  43. [50]

    Proof of an entropy conjecture for Bloch coherent spin states and its generalizations

    Elliott H. Lieb and Jan Philip Solovej. “Proof of an entropy conjecture for Bloch coherent spin states and its generalizations”. In:Acta Math.212.2 (2014), pp. 379–398

    2014

  44. [51]

    The weak convergence of varifolds gener- ated by rectifiable flatG-chains

    Chunyan Liu, Yangqin Fang, and Ning Zhang. “The weak convergence of varifolds gener- ated by rectifiable flatG-chains”. In:Bull. Lond. Math. Soc.54.6 (2022), pp. 2351–2371

    2022

  45. [53]

    Zhenhua Liu.Area-minimizing surfaces with fractal singular sets: prevalence, moduli space, and refinements on strata. 2023. arXiv:2306.07271 [math.DG]

    2023 arXiv

  46. [54]

    accepted at Annals of Mathematics

    Zhenhua Liu.On a conjecture of Almgren: area-minimizing submanifolds with fractal singularities. accepted at Annals of Mathematics. arXiv:2110.13137 [math.DG]

  47. [55]

    Zhenhua Liu.The Hasse Principle for Geometric Variational Problems: An Illustration via Area- minimizing Submanifolds. 2026. arXiv:2508.21045 [math.DG]

    2026

  48. [56]

    Connected sums of codimension two locally flat submanifolds

    Charles Livingston. “Connected sums of codimension two locally flat submanifolds”. In: Proceedings of the Royal Society of Edinburgh: Section A Mathematics(2023), 1–8

    2023

  49. [57]

    A priori bounds for geodesic diameter. Part I. Integral chains with coefficients in a complete normed commutative group

    Ulrich Menne and Christian Scharrer. “A priori bounds for geodesic diameter. Part I. Integral chains with coefficients in a complete normed commutative group”. In:Rev. Mat. Iberoam. 41.1 (2025), pp. 29–72

    2025

  50. [58]

    Calibrations moduloν

    Frank Morgan. “Calibrations moduloν”. In:Adv. in Math.64.1 (1987), pp. 32–50

    1987

  51. [59]

    Examples of unoriented area-minimizing surfaces

    Frank Morgan. “Examples of unoriented area-minimizing surfaces”. In:Trans. Amer. Math. Soc.283.1 (1984), pp. 225–237

    1984

  52. [60]

    Twisted calibrations

    Timothy A. Murdoch. “Twisted calibrations”. In:Trans. Amer. Math. Soc.328.1 (1991), pp. 239– 257

    1991

  53. [61]

    Thesis (Ph.D.)–Rice University

    Timothy Armstrong Murdoch.Twisted-calibrations and the cone on the Veronese surface. Thesis (Ph.D.)–Rice University. ProQuest LLC, Ann Arbor, MI, 1988, p. 48

    1988

  54. [62]

    Cylindrical tangent cones and the singular set of minimal submanifolds

    Leon Simon. “Cylindrical tangent cones and the singular set of minimal submanifolds”. In: J. Differential Geom.38.3 (1993), pp. 585–652

    1993

  55. [63]

    Leon Simon.Lectures on geometric measure theory. Vol. 3. Proceedings of the Centre for Mathe- matical Analysis, Australian National University. Australian National University, Centre for Mathematical Analysis, Canberra, 1983, pp. vii+272

    1983

  56. [64]

    Generic regularity of homologically area minimizing hypersurfaces in eight- dimensional manifolds

    Nathan Smale. “Generic regularity of homologically area minimizing hypersurfaces in eight- dimensional manifolds”. In:Comm. Anal. Geom.1.2 (1993), pp. 217–228

    1993

  57. [65]

    Minimizing cones associated with isoparametric foli- ations

    Zizhou Tang and Yongsheng Zhang. “Minimizing cones associated with isoparametric foli- ations”. In:J. Differential Geom.115.2 (2020), pp. 367–393

    2020

  58. [66]

    Quelques propri´et´es globales des vari´et´es diff´erentiables

    Ren ´e Thom. “Quelques propri´et´es globales des vari´et´es diff´erentiables”. In:Comment. Math. Helv.28 (1954), pp. 17–86. REFERENCES 49

    1954

  59. [67]

    Stratification theory

    David Trotman. “Stratification theory”. In:Handbook of geometry and topology of singularities. I. Springer, Cham, [2020] ©2020, pp. 243–273

    2020

  60. [68]

    Geometric versions of Whitney regularity for smooth stratifications

    David J. A. Trotman. “Geometric versions of Whitney regularity for smooth stratifications”. In:Ann. Sci. ´Ecole Norm. Sup. (4)12.4 (1979), pp. 453–463

    1979

  61. [69]

    Mean convex smoothing of mean convex cones

    Zhihan Wang. “Mean convex smoothing of mean convex cones”. In:Geom. Funct. Anal.34.1 (2024), pp. 263–301

    2024

  62. [70]

    Generic Transversality of Minimal Submanifolds and Generic Regularity of Two-Dimensional Area-Minimizing Integral Currents

    Brian White. “Generic Transversality of Minimal Submanifolds and Generic Regularity of Two-Dimensional Area-Minimizing Integral Currents”. In: (). arXiv:1901.05148 [math.DG]

    1901 arXiv

  63. [71]

    Rectifiability of flat chains

    Brian White. “Rectifiability of flat chains”. In:Ann. of Math. (2)150.1 (1999), pp. 165–184

    1999

  64. [72]

    The deformation theorem for flat chains

    Brian White. “The deformation theorem for flat chains”. In:Acta Math.183.2 (1999), pp. 255– 271

    1999

  65. [73]

    Thesis (Ph.D.)–State University of New York at Stony Brook

    Yongsheng Zhang.Gluing Techniques in Calibrated Geometry. Thesis (Ph.D.)–State University of New York at Stony Brook. ProQuest LLC, Ann Arbor, MI, 2013, p. 97

    2013

  66. [74]

    On extending calibration pairs

    Yongsheng Zhang. “On extending calibration pairs”. In:Adv. Math.308 (2017), pp. 645–670

    2017

  67. [75]

    On realization of tangent cones of homologically area-minimizing com- pact singular submanifolds

    Yongsheng Zhang. “On realization of tangent cones of homologically area-minimizing com- pact singular submanifolds”. In:J. Differential Geom.109.1 (2018), pp. 177–188. I n [ 1 ] : = (*First let us define v_1,...,v_5 and verify they form an orthonormal set*) v[1]:= 1 Sqrt[3] 1 ...

    2018

  68. [2026]

    arXiv:2304.10653 [math.DG]

    arXiv

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