Pith. sign in

REVIEW 3 major objections 5 minor 35 references

Hausdorff measure bounds for density-$Q$ flat singularities of minimizing integral currents

T0 review · 3 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper proves that flat singular points of locally highest density of area-minimizing integral currents have locally finite (m−2)-dimensional Hausdorff measure.

desk verdict The paper announces a major measure-bound result but defers the central Minkowski-content proof; referee it seriously and make acceptance contingent on closing that gap. read the letter →

arxiv 2504.19234 v1 pith:PXW5NWOP submitted 2025-04-27 math.DG math.AP

Gianmarco Caldini , Anna Skorobogatova This is my paper · ORCID
classification math.DGmath.AP MSC 49Q1549Q2028A78
keywords integralcurrentsarea-minimizingflatsingularitiessingularitydegreeHausdorffmeasureMinkowskicontentfrequencyfunctioncentermanifold
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

This paper proves that, for an area-minimizing integral current of dimension $m$ in a smooth Riemannian manifold, the set of flat singular points of a fixed locally maximal density $Q$ has locally finite $(m-2)$-dimensional Hausdorff measure, i.e. $H^{m-2}(F_Q(T)) < \infty$. The proof splits $F_Q(T)$ into two parts using the singularity degree $I(T,p)$. Points with singularity degree at most $1+\delta$ form an $H^{m-2}$-negligible set, while points with singularity degree at least $1+\delta$ have finite $(m-2)$-dimensional Minkowski content. Together these two estimates give the finite-measure conclusion and provide a quantitative size bound for the top-density flat part of the singular set, complementing the known rectifiability of the singular set.

What carries the argument

The central object is the singularity degree $I(T,p)$, defined as the infimum of the frequency values of fine blow-up limits at $p$; it measures how homogeneous the first nontrivial blow-up term is. The proof is carried by three mechanisms. Proposition 3.1 is a uniform tilt-excess decay: at points with $I(T,p) \ge I_0 > 1$, the tilt excess $E(T,B_r(p))$ decays like $(r/r_0)^\alpha$, with the threshold $r_0$ depending only on $I_0$, $m$, $n$, $Q$ and not on $T$ or $p$. Lemma 4.2 is a quantitative dichotomy: at low-degree points the current is either close to a multi-plane cone with a common $(m-2)$-dimensional spine, or all density-$Q$ points are trapped near an $(m-3)$-dimensional subspace; this feeds the conical excess decay theorem used to prove $H^{m-2}$-negligibility. For high-degree points, a $(1+\delta)$-stopping and restarting procedure constructs center manifolds and intervals of flattening over the whole set $F_{Q,\ge 1+\delta}(T)$ at once, avoiding the countable decomposition of earlier work, and yields the $\beta_2$-coefficient and iterative covering estimates behind the Minkowski content bound.

What would settle it

One could look for a sequence of scales $r_j\to 0$ and centers $x_j\in F_{Q,\ge 1+\delta}(T)$ for which the single-piece covering estimate (2) fails, e.g. $|B_{r_j}(F_{Q,\ge 1+\delta}(T))| \ge c r_j^{n+2-\varepsilon}$ for some $\varepsilon>0$ while each point still satisfies the uniform tilt-excess decay of Proposition 3.1; such a current would violate the finite Minkowski content conclusion of Theorem 1.4.

Watch

Extended reading notes

Core claim

Under Assumption 1.1, Theorem 1.2 asserts $H^{m-2}(F_Q(T)) < \infty$ for the flat singular points of density $Q$. The argument establishes two stronger statements: Theorem 1.5 shows $H^{m-2}(F_{Q,\le 1+\delta}(T)) = 0$ for every $\delta \in (0,1/Q)$, and Theorem 1.4 shows that $F_{Q,\ge 1+\delta}(T)$ has finite $(m-2)$-dimensional upper Minkowski content, meaning $|B_r(F_{Q,\ge 1+\delta}(T))| \le C r^{n+2}$ for all small $r$ with a constant independent of the single point. The low-degree part is handled by a quantitative version of the frequency-one analysis, and the high-degree part is handled by applying the covering construction to the entire set at once rather than to countably many pieces.

Load-bearing premise

The load-bearing premise of the Minkowski-content half is that the center-manifold and covering construction, applied to the whole set $F_{Q,\ge 1+\delta}(T)$ as a single piece, still yields the $\beta_2$-coefficient estimates and iterative covering bounds that the original construction yields piecewise; the paper sets this up in Section 5 and defers the full proof to [11] and [26, Part 2].

Editorial extensions

If this is right

  • If the theorem is right, every current satisfying Assumption 1.1 has a top-density flat singular set with finite $(m-2)$-dimensional Hausdorff measure, not merely a rectifiable one of possibly infinite measure.
  • The low-degree set $F_{Q,\le 1+\delta}(T)$ is $H^{m-2}$-negligible, so the entire $(m-2)$-measure of $F_Q(T)$ is carried by points with singularity degree at least $1+\delta$.
  • The high-degree set has finite $(m-2)$-dimensional Minkowski content, which implies finite upper Minkowski dimension at most $m-2$ and is a stronger quantitative control than Hausdorff measure alone.
  • At $H^{m-2}$-almost every flat singular point of density $Q$, the singularity degree is at least $1+1/Q$.
  • The same two-part argument adapts to integral currents semicalibrated by a smooth differential form, upgrading the known rectifiability result to local Hausdorff measure bounds for density-$Q$ flat singularities.

Reading between the lines

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

  • The local statement should globalize to compact ambient manifolds by a finite covering argument, giving a global finite $(m-2)$-Hausdorff bound for top-density flat singularities without new analytic input.
  • A natural next target is the remaining set $F_{Q,\le 1+\delta}(T)$: a Minkowski-content bound there would upgrade Theorem 1.2 to a full Minkowski-content statement for all density-$Q$ flat singularities, and the paper's own discussion points to the 'holes' condition in the covering argument as the obstruction.
  • The uniformity in Proposition 3.1 suggests the constant in (1) depends only on $Q$, $m$, $n$ and the ambient geometry, so one could attempt to extract explicit rates from the deferred proofs and test them on model examples such as unions of complex planes in $\mathbb{C}^n$.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper studies the singular set of area-minimizing integral currents of dimension m and general codimension in a smooth Riemannian manifold. Its main theorem, Theorem 1.2, asserts that the set F_Q(T) of flat singular points of locally highest density Q has locally finite (m-2)-dimensional Hausdorff measure. The proof splits F_Q(T) into two pieces: F_{Q,<=1+delta}(T), for which Theorem 1.5 claims H^{m-2}-nullity, and F_{Q,>=1+delta}(T), for which Theorem 1.4 claims finite (m-2)-dimensional Minkowski content. The main new ingredients are a uniform tilt-excess decay (Proposition 3.1), a quantitative version of a dichotomy lemma from the authors' prior work (Lemma 4.2), and a single-piece adaptation of the center-manifold construction of [11] and [26] to avoid the countable frequency decomposition that prevented Minkowski content bounds. The paper is largely a research announcement: several key steps are summarized with explicit references to [9], [10], [11], [26], and [31] for the full arguments.

Significance. If Theorem 1.2 is fully established, it is a substantial advance: it upgrades the known (m-2)-rectifiability of flat singularities of top density to a quantitative Hausdorff measure bound, and the uniform-in-center tilt excess decay of Proposition 3.1 is a genuine strengthening of [10, Proposition 7.2]. The paper is transparent about its main obstruction and about what is deferred, which is a strength. The central claim is not circular: the authors do not assume the conclusion, and the theorems are not equivalent to any single cited result. However, the proof of Theorem 1.4, which is load-bearing for Theorem 1.2, is not contained in the manuscript: Section 5 describes a construction and then refers to [11] and [26, Part 2] for the full proof. This is a verifiability gap rather than an evident contradiction, but it prevents the current version from being accepted as a complete proof.

major comments (3)
  1. [Section 5, Theorem 1.4 and Eq. (2)] The proof of the Minkowski content bound (2) is not present in the manuscript. After defining the single piece S = S_{K0} and stating the setup with adapted center manifolds, the section lists four steps (1)-(4) and then says 'we refer the reader to [11], [26, Part 2] for the full proof.' The crucial new point is precisely that the countable frequency decomposition of [11] is replaced by the entire set S as a single piece. The manuscript does not prove that the beta-2 coefficient estimates of [11, Proposition 13.2] and the iterative covering argument of [11, Appendix A] hold for this single, not-necessarily-closed set with constants uniform over all frequency bands. Since Theorem 1.4 is one of the two pillars of Theorem 1.2, this is a load-bearing gap. The authors should either supply the complete adaptation, or state precisely which estimates from [11] and [26, Part 2] are being invoked and why they pass through unchanged in the single-piece setting.
  2. [Section 3.1, Proposition 3.3] Proposition 3.3 is the compactness result on which Proposition 3.1 depends, and its proof is asserted in one sentence: it is said to 'follow verbatim' from [10, Proposition 4.1] with [10, Lemma 4.5] unchanged in the setting of varying blowup centers. The uniformity of the radius r_0(I_0,m,n,Q) in Proposition 3.1 is a key novelty of the paper and is used essentially in Section 5. The manuscript does not explain how the varying-center compactness argument, the condition (9), and the conclusion v = lambda u are obtained without changes. At minimum, the exact modifications to [10, Lemma 4.5] should be described so that the uniformity claim can be checked.
  3. [Section 4, Theorem 1.5] The transition from Lemma 4.2 to the H^{m-2}-nullity of F_{Q,<=1+delta}(T) is announced as an immediate corollary, but the covering argument is not written out. Lemma 4.2 provides a scale-dependent dichotomy, with alternative (b) involving an (m-3)-dimensional affine subspace, and the manuscript does not show how the p-dependent scale rho(p,epsilon) and the quantitative constants combine with Proposition 3.1 to yield a global null-set estimate for every delta < 1/Q. Since Theorem 1.5 is the second pillar of Theorem 1.2, the details should be included or the precise statement in [9] or [31] that supplies the covering should be identified, together with the modification needed for the range of singularity degrees considered here.
minor comments (5)
  1. [Assumption 1.0] The line 'n ≥ n ≥ 2' contains a typo; it should presumably be 'n ≥ 2'. The dimensional notation around R^{m+n} = R^{m+n+l} is also confusing and should be clarified.
  2. [Theorem 1.4] The statement 'there exists r_1 = r_1(m,n,Q,delta) such that and C(Q,m,n,delta)>0 such that' is grammatically malformed; it should read 'there exist r_1 > 0 and C > 0 such that'.
  3. [Proof of Theorem 1.2] In the proof of Theorem 1.2, the reference to 'Proposition 1.5' should be to Theorem 1.5.
  4. [Theorem 4.1 and references] The constants in Theorem 4.1 are written as C(Q,m,n,n), which appears to be a typo for C(Q,m,n) or a similar expression. Also, in the bibliography, reference [25] contains the typo 'extimates' for 'estimates'.
  5. [Notation in Section 5] The bold notation 'mmm_{x,k}' in Eq. (21) is nonstandard and is confusing; it should be replaced by a clearly defined symbol such as m_{x,k} or M_{x,k}.

Circularity Check

1 steps flagged · score 4.0 of 10

Theorem 1.4 is deferred to the authors' own preprints; the load-bearing single-piece Minkowski-content estimate is cited, not proved.

  1. self citation load bearing [Section 5, after the definition of S_K0 and before the proof of Theorem 1.4]
    "For the purpose of clarity, we provide the setup and describe the key ideas here and we refer the reader to [11], [26, Part 2] for the full proof. ... Therein such a refined decomposition procedure is not necessary and is merely used for convenience, whereas here it is crucial in order to obtain the content bound (2)."

    The entire content of Theorem 1.4 is the single-piece Minkowski-content bound (2) for F_{Q,≥1+δ}(T). The manuscript does not prove this estimate: Section 5 defines S_K0, asserts that Proposition 3.1 lets one work with this single piece, lists four ingredients from [11], and concludes by referring to [11] and [26, Part 2]. But [11] works with the countable pieces S_K or S-tilde_K and the paper itself says the [26] adaptation was 'merely used for convenience' there. Thus the crucial step — that the single-piece S_K0 construction yields the uniform, closed-set covering needed for a Minkowski-content bound — is justified only by the authors' own unverified preprints, not by an exhibited derivation or an independent theorem.

full rationale

The derivation of Theorem 1.2 is not self-definitional: F_Q(T), the singularity degree, and the Hausdorff measure are independent objects, and Theorems 1.4 and 1.5 are not restatements of Assumption 1.1. There is also no fitted parameter renamed as a prediction, so the definitional and fitted-input categories do not apply. The paper contains substantial independent arguments: Proposition 3.1 is developed with a detailed contradiction scheme, and Section 4 gives a substantial proof of Lemma 4.2 leading to Theorem 1.5. However, the second leg of the split — Theorem 1.4, the finite Minkowski-content bound for F_{Q,≥1+δ}(T) — is not proved in the manuscript. Section 5 defines the single set S_K0, asserts that Proposition 3.1 lets one avoid the countable decomposition of [11], lists four ingredients from [11], and then says the full proof is in [11] and [26, Part 2]. Since [11] and [26] are preprints by the same research group, and the paper itself says the [26] adaptation was 'merely used for convenience' there, this is a load-bearing self-citation rather than independent support. The central claim therefore rests partly on an unverified self-citation chain, but not on a definitional or fitted-input equivalence, so the score is moderate rather than high.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the standard regularity framework for area-minimizing currents (Almgren, De Lellis-Spadaro) and on several theorems from the authors' own prior work. No data-fitted parameters or invented entities appear; proof constants such as ε, δ, κ are existential choices within the arguments, not free parameters.

assumptions (5)
  • domain assumption Assumption 1.0: Σ is C^{3,κ0} with c(Σ) ≤ ε small
    Localized ambient manifold and small curvature are required for the center manifold and approximation theorems used throughout; the authors note a possible weakening to C^{2,κ0} via [28] but do not pursue it.
  • domain assumption Assumption 1.1: 0 is a flat singular point with density Q and (p_{π0})♯ T restricted to B_{6√m} equals Q times the plane
    This normalization expresses the 'locally highest density' and flatness condition on which the decomposition into F_{Q,≤1+δ} and F_{Q,≥1+δ} rests.
  • standard math The center manifold, Lipschitz approximation, and height bound theorems of De Lellis-Spadaro ([13], [15], [16]) and Spolaor ([34])
    Used in the compactness procedure, the definition of flattening intervals, and the blow-up analysis in Sections 2 and 3.
  • standard math The classification of 1-homogeneous Q-valued Dir-minimizers in two dimensions ([12, Proposition 5.1])
    Used in Lemma 4.2 to force the singularity degree to be exactly 1 when the coarse blow-up is (m-2)-invariant.
  • standard math The conical excess decay theorem of De Lellis-Minter-Skorobogatova ([9, Theorem 2.5])
    Applied in Section 4 to rescaled currents at flat singular points of degree below 1+1/Q.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Hausdorff measure bounds for density-$Q$ flat singularities of minimizing integral currents." pith.science (2026). https://pith.science/paper/PXW5NWOP

@misc{pith2026250419234,
  author       = {Pith},
  title        = {Pith review of: Hausdorff measure bounds for density-$Q$ flat singularities of minimizing integral currents},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PXW5NWOP}},
  note         = {Machine review of arXiv:2504.19234}
}
abstract

In this article we prove that the set of flat singular points of locally highest density of area-minimizing integral currents of dimension $m$ and general codimension in a smooth Riemannian manifold $\Sigma$ has locally finite $(m-2)$-dimensional Hausdorff measure. In fact, the set of such flat singular points can be split into a union of two sets, one of which we show is locally $\mathcal{H}^{m-2}$-negligible, while for the other we obtain local $(m-2)$-dimensional Minkowski content bounds.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

35 extracted references · 30 canonical work pages

  1. [11]

    De Lellis and A

    C. De Lellis and A. Skorobogatova,The fine structure of the singular set of area-minimizing integral currents II: rectifiability of flat singular points with singularity degree larger than 1, preprint arXiv:2304.11555

    arXiv

  2. [26]

    Minter, D

    P. Minter, D. Parise, A. Skorobogatova and L. Spolaor,Rectifiability of the singular set and uniqueness of tangent cones for semicalibrated currents, preprint, arXiv:2409.03037

    arXiv

  3. [9]

    De Lellis, P

    C. De Lellis, P. Minter and A. Skorobogatova,The Fine Structure of the Singular Set of Area-Minimizing Integral Currents III: Frequency 1 Flat Singular Points and Hm−2 -a.e. Uniqueness of Tangent Cones, preprint arXiv:2304.11553

    arXiv

  4. [10]

    Ars Inveniendi Analytica

    C. De Lellis and A. Skorobogatova,The fine structure of the singular set of area-minimizing integral currents I: the singularity degree of flat singular points, in “Ars Inveniendi Analytica", 3, 2025

    work page 2025

  5. [31]

    Journal of Differential Geometry

    L. Simon,Cylindrical tangent cones and the singular set of minimal submanifolds, in “Journal of Differential Geometry",38(3), 585–652, 1993

    work page 1993

  6. [1]

    Annals of Mathematics

    W. Allard,On the first variation of a varifold, in “Annals of Mathematics",95(3), 417–491, 1972

    work page 1972

  7. [2]

    Annals of Mathematics

    F. Almgren,Some interior regularity theorems for minimal surfaces and an extension of Bernstein’s theorem, in “Annals of Mathematics",84(2), 277–292, 1966

    work page 1966

  8. [3]

    Bulletin of the American Mathematical Society (N.S.)

    F. J. Almgren,Q-valued functions minimizing Dirichlet’s integral and the regularity of area minimizing rectifiable currents up to codimension two, in “Bulletin of the American Mathematical Society (N.S.)",8(2), 327–328, 1983

    work page 1983

Show all 35 references
  1. [4]

    Almgren,Almgren’s Big Regularity Paper, World Scientific Monograph Series in Mathematics,1, World Scientific Publishing Co

    F. Almgren,Almgren’s Big Regularity Paper, World Scientific Monograph Series in Mathematics,1, World Scientific Publishing Co. Inc., River Edge 2000

    2000

  2. [5]

    Inventiones Mathematicae

    E. Bombieri, E. De Giorgi and E. Giusti,Minimal cones and the Bernstein problem, in “Inventiones Mathematicae",7, 243–268, 1969

    1969

  3. [6]

    Journal of the American Mathematical Society

    S. Chang,Two-dimensional area minimizing integral currents are classical minimal surfaces, in “Journal of the American Mathematical Society",1(4), 699–778, 1988

    1988

  4. [7]

    De Giorgi,Frontiere orientate di misura minima, inSeminario di Matematica della Scuola Normale Superiore di Pisa A.A

    E. De Giorgi,Frontiere orientate di misura minima, inSeminario di Matematica della Scuola Normale Superiore di Pisa A.A. 1960-1961, Editrice Tecnico Scientifica, Pisa 1961, 1–56

    1960

  5. [8]

    Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

    E. De Giorgi,Una estensione del teorema di Bernstein, in “Annali della Scuola Normale Superiore di Pisa - Classe di Scienze",19(3), 79–85, 1965

    1965

  6. [12]

    Memoirs of the American Mathematical Society

    C. De Lellis and E. Spadaro,Q-valued functions revisited, in “Memoirs of the American Mathematical Society",211(991), 2011

    2011

  7. [13]

    Geometric and Functional Analysis

    C. De Lellis and E. Spadaro,Regularity of area minimizing currents I: gradient L p estimates, in “Geometric and Functional Analysis",24, 1831–1884, 2014

    2014

  8. [14]

    Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

    C. De Lellis and E. Spadaro,Multiple valued functions and integral currents, in “Annali della Scuola Normale Superiore di Pisa - Classe di Scienze",14(5), 1239–1269, 2015

    2015

  9. [15]

    Annals of Mathematics

    C. De Lellis and E. Spadaro,Regularity of area minimizing currents II: center manifold, in “Annals of Mathematics",183(2), 499–575, 2016

    2016

  10. [16]

    Annals of Mathematics

    C. De Lellis and E. Spadaro,Regularity of area minimizing currents III: blow-up, in “Annals of Mathematics",183(2), 577–617, 2016. 24Gianmarco Caldini and Anna Skorobogatova

    2016

  11. [17]

    Annals of PDE

    C. De Lellis, E. Spadaro and L. Spolaor,Regularity theory for2-dimensional almost minimal currents II: Branched center manifold, in “Annals of PDE",3, 18, 2017

    2017

  12. [18]

    Transactions of the American Mathematical Society

    C. De Lellis, E. Spadaro and L. Spolaor,Regularity theory for2-dimensional almost minimal currents I: Lipschitz approximation, in “Transactions of the American Mathematical Society", 370(3), 1783–1801, 2018

    2018

  13. [19]

    Journal of Differential Geometry

    C. De Lellis, E. Spadaro and L. Spolaor,Regularity theory for2-dimensional almost minimal currents III: Blowup, in “Journal of Differential Geometry",116(1), 125–185, 2020

    2020

  14. [20]

    Bulletin of the American Mathematical Society

    H. Federer,The singular sets of area minimizing rectifiable currents with codimension one and of area minimizing flat chains modulo two with arbitrary codimension, in “Bulletin of the American Mathematical Society",76(4), 767–771, 1970

    1970

  15. [21]

    Federer,Geometric measure theory, Classics in Mathematics, Springer, Berlin-Heidelberg 1996

    H. Federer,Geometric measure theory, Classics in Mathematics, Springer, Berlin-Heidelberg 1996

    1996

  16. [22]

    Rendiconti del Circolo Matematico di Palermo

    W. Fleming,On the oriented Plateau problem, in “Rendiconti del Circolo Matematico di Palermo",11(1), 69–90, 1962

    1962

  17. [23]

    Krummel and N

    B. Krummel and N. Wickramasekera,Analysis of singularities of area minimizing currents: a uniform height bound, estimates away from branch points of rapid decay, and uniqueness of tangent cones, preprint, arXiv:2304.10272

    arXiv

  18. [24]

    Krummel and N

    B. Krummel and N. Wickramasekera,Analysis of singularities of area minimizing currents: planar frequency, branch points of rapid decay, and weak locally uniform approximation, preprint, arXiv:2304.10653

    arXiv

  19. [25]

    Krummel and N

    B. Krummel and N. Wickramasekera,Analysis of singularities of area minimising currents: higher order decay extimates at branch points and size and rectifiability of the singular set, in preparation

  20. [27]

    Journal of the European Mathematical Society

    A. Naber and D. Valtorta,The singular structure and regularity of stationary varifolds, in “Journal of the European Mathematical Society",22(10), 3305–3382, 2020

    2020

  21. [28]

    Nardulli and R

    S. Nardulli and R. Resende,Interior regularity of area minimizing currents within a C2,α -submanifold, preprint, arXiv:2404.17407

    arXiv

  22. [29]

    Simon,Lectures on geometric measure theory, Proceedings of the Centre for Mathematical Analysis,3, Australian National University, Canberra 1984

    L. Simon,Lectures on geometric measure theory, Proceedings of the Centre for Mathematical Analysis,3, Australian National University, Canberra 1984

    1984

  23. [30]

    Surveys in Differential Geometry

    L. Simon,Rectifiability of the singular sets of multiplicity 1 minimal surfaces and energy minimizing maps, proceedings of the Conference on Geometry and Topology (Cambridge, MA 23-25 April, 1993), in “Surveys in Differential Geometry",2, 246-305, 1995

    1993

  24. [32]

    Annals of Mathematics

    J. Simons,Minimal varieties in Riemannian manifolds, in “Annals of Mathematics",88(1), 62–105, 1968. Hausdorff measure bounds for density-Q flat singularities of minimizing integral currents25

    1968

  25. [33]

    Communications on Pure and Applied Mathematics

    A. Skorobogatova,An upper Minkowski dimension estimate for the interior singular set of area minimizing currents, in “Communications on Pure and Applied Mathematics",77(2), 1509–1572, 2024

    2024

  26. [34]

    Advances in Mathe- matics

    L. Spolaor,Almgren’s type regularity for semicalibrated currents, in “Advances in Mathe- matics",350, 749–815, 2019

    2019

  27. [35]

    Journal f¨ur die reine und angewandte Mathematik

    B. White,Stratification of minimal surfaces, mean curvature flows, and harmonic maps, in “Journal f¨ur die reine und angewandte Mathematik",488, 1–35, 1997. DEPARTMENT OFMATHEMATICS, UNIVERSIT`A DITRENTO, 38123 TRENTO, ITALY. E-mail address:gianmarco.caldini@unitn.it INSTITUTE...

    1997

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.