Regularity of branched stable minimal immersed hypersurfaces

Gaoming Wang; Xuwen Zhang

REVIEW 3 major objections 4 minor 27 references

Regularity of branched stable minimal immersed hypersurfaces

T0 review · 3 major / 4 minor · reviewed 2026-08-03 · deepseek-v4-flash

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

Pith's one-line read The paper proves a sharp Hausdorff-dimension bound for non-branch singular sets of branched stable minimal immersed hypersurfaces, and constructs examples attaining it in every dimension.

desk verdict Sharp non-branch singularity bound for branched stable minimal immersions; probably right, but the central ε-regularity step is imported from Bellettini by reference and needs to be written out. read the letter →

arxiv 2607.29632 v1 pith:QZVSXQM6 submitted 2026-07-31 math.DG math.AP

Gaoming Wang , Xuwen Zhang This is my paper

classification math.DGmath.AP MSC 49Q1549Q2053A1053C42

keywords branchedminimalimmersionsstablehypersurfacesnon-branchsingularsetHausdorffdimensionvarifoldcompactnesssheetingtheoremtiltfunctioncones

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

The paper establishes that when limits of stable minimal immersed hypersurfaces are allowed to branch, the only genuinely singular points left over after removing branch points form a very small set: empty in two dimensions, isolated in three dimensions, and of Hausdorff dimension at most n−3 in higher dimensions. This is the immersed analogue of the sharp regularity known for embedded stable hypersurfaces, now extended to settings where branching is possible. The proof introduces a tilt function that detects the normal directions of a limiting cone, converting stability into a differential inequality that forces the hypersurface to decompose into Lipschitz multi-valued graphs. The authors also construct stable minimal cones showing the n−3 bound is attained in every dimension, so the result is optimal. This matters because it isolates the local analysis near classical cones as the missing ingredient, and reduces the remaining compactness question to the structure of branch sets of multi-valued graphs.

What carries the argument

The key object is a tilt function g_p = G_p ∘ ν, built from the normals of the cone. G_p is chosen as sqrt(1 − |π_L(y)|² + k ∏ φ(1 − ⟨y, p_i⟩)), where the p_i are the cone's normal directions, L is their span, and φ vanishes only at 0. The generalized differential inequality |A|² g_p² + g_p Δg_p ≥ C_0 |A|² turns stability of the hypersurface into a weighted energy inequality for g_p. That inequality drives an iteration argument which yields the branched sheeting theorem: a stable minimal immersion close to such a cone is a superposition of Lipschitz multi-valued graphs over the cone's planes, with Lipschitz constants controlled by the tilt energy. The sharpness examples use a polar-map const

What would settle it

Run the spectral calculation for the family of high-genus minimal surfaces used in Section 6: if the first eigenvalue is at most 2 infinitely often, the example in Theorem 1.5 collapses. Alternatively, construct a varifold limit of stable branched minimal immersed hypersurfaces whose non-branch singular set has Hausdorff dimension n−2; that would directly contradict the bound n−3.

Watch

Extended reading notes

Core claim

The central claim is that branching is the only source of large singular sets in limits of stable minimal immersed hypersurfaces. For a varifold limit V of such hypersurfaces whose singular sets have locally finite (n−2)-dimensional measure, the non-branch singular set — points where no tangent cone is a sum of hyperplanes — is empty when n=2, discrete when n=3, and has Hausdorff dimension at most n−3 when n≥4. The proof works through a branched sheeting theorem: near a stationary classical cone or hyperplane cone whose spine has dimension at least n−2, any stable minimal immersion with small tilt energy and small varifold distance must decompose into Lipschitz multi-valued graphs over the c

Load-bearing premise

The sharpness construction rests on a spectral estimate for the desingularized tori: the first eigenvalue of −Δ + |A|²/8 must stay above 2 as the genus grows; if that gap fails, the stable cone with a genuine non-branch singularity may not exist, and the claimed sharpness in every dimension loses its witness (the dimension bound itself could still hold).

Editorial extensions

If this is right

Varifold limits of stable branched minimal immersed hypersurfaces with locally finite H^(n−2) singular sets remain in the same class, and their non-branch singular sets obey the stated dimension bounds.
In dimension 2 there are no non-branch singular points in such limits; in dimension 3 such points form at most a discrete set.
The product examples give stable minimal cones in every dimension n≥3 whose non-branch singular set has Hausdorff dimension exactly n−3, so the exponent in the main theorem cannot be lowered.
The branched sheeting theorem provides Lipschitz multi-valued graphical decompositions near hyperplane and classical cones, with slope control in terms of tilt energy, which is the local ingredient needed for the compactness program.
The remaining obstacle to the full compactness conjecture is a structure theorem for the branch set of stationary Lipschitz multi-valued graphs; the paper isolates this as the only missing piece.

Reading between the lines

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

Editorial inference: The same tilt-function ansatz may extend to cones with normal rank at least 4, which would replace the spine-dimension condition by a more flexible criterion; if the algebraic lower bounds fail there, the n−3 exponent may be tied to the method rather than to the geometry.
Editorial inference: The spectral criterion (first eigenvalue of −Δ + |A|²/8 exceeding 2) is a testable recipe for generating other stable branched minimal cones; varying the desingularized surfaces could yield cones with intermediate dimensions of non-branch singular sets, not just the extremal examples.
Editorial inference: The n−3 bound is four dimensions larger than the codimension-7 estimates known from embedded and non-branched theories, suggesting that allowing branch points is what forces the larger singular set; the open branch-set structure question will decide whether this deficit is realized in examples.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

This is the sharp non-branch regularity bound for branched stable minimal immersed hypersurfaces, and I think the main theorem is right — but the paper's central ε-regularity step is imported from Bellettini by reference rather than proved, and a referee should insist it be written out.

What is actually new is substantial. Theorem 1.3 — non-branch singular set empty for n=2, discrete for n=3, Hausdorff dimension at most n−3 for n≥4, plus precompactness — is the natural optimal statement in Bellettini's program, and the architecture is right: a generalized Schoen inequality (Theorem 1.7) for cones of normal rank 2 or 3, then a branched sheeting theorem (Theorem 1.8), then dimension reduction. Section 3 is real work, not a routine adaptation: the tilt function with the product ansatz, the algebraic trace lemmas, and the near-zero / away-from-zeros / transition trichotomy are carried out in detail. The sharpness construction is also honest. It depends on the Kapouleas–Wiygul gluing estimates, but the dependence is stated precisely, and Appendix B contains an actual spectral argument — the Dirichlet limit on the four cylinders gives 17/4, comfortably above the 2 needed for stability — rather than a hand-wave.

The soft spots. The stress-test concern lands, with a qualifier. Theorem 4.3 is dispatched by 'one can follow essentially the same argument as in [Bel25]'. Bellettini's iteration was written for a tilt function with one zero; here there are finitely many normals and the superlevel sets may cross the H^{n−2}-finite singular set. What needs checking — level-independent Sobolev constants, the precise role of L and \tilde C, the branch-set crossing — is not in the text. My own read is that the translation is routine: Proposition 4.2 gives the Caccioppoli inequalities, the ambient Sobolev inequality doesn't see the branch set, and |∇g| ≤ C|A| is proved in full. So I would bet the import works, and I'm more optimistic than the LOW confidence in the reader's report suggests. But 'routine' is exactly what the paper should demonstrate, since everything downstream (Theorem 1.8, and through it Theorem 1.3) rides on it. Secondary issues: the d_i = q_i step in Theorem 1.8 is compressed, and the paper never states why the sharpness cones belong to the closure class V̄(Λ) — presumably standard, but worth one sentence. Neither looks load-bearing.

For whom: GMT and minimal-surface regularity specialists. It deserves a serious referee — send it, with a request for the Theorem 4.3 expansion.

Referee Report

3 major / 4 minor

Summary. The paper establishes a sharp Hausdorff-dimension bound for the non-branch singular set of branched stable minimal immersed hypersurfaces whose singular sets have locally finite H^{n-2}-measure: the non-branch singular set is empty for n=2, discrete for n=3, and has Hausdorff dimension at most n-3 for n>=4 (Theorem 1.3). The proof introduces a generalized Schoen differential inequality for cones of normal rank 2 or 3 (Theorem 1.7) and a branched sheeting theorem near such cones (Theorem 1.8), then applies a Federer dimension-reduction argument. The paper also constructs stable minimal cones with non-branch singularities at the vertex (Theorem 1.5), using Kapouleas-Wiygul desingularizations and Lawson's polar map, to demonstrate sharpness in every dimension.

Significance. If the proof is complete, this is a major advance: it supplies the missing local analysis near stationary classical cones identified by Bellettini as the key step toward a compactness theory for branched stable minimal immersed hypersurfaces. The tilt-function construction for cones with normal rank 2 and 3 (Theorem 1.7) is a significant technical achievement and is carried out in substantial detail. The Caccioppoli inequalities in Section 4 are proved explicitly. The sharpness example, though dependent on the external gluing construction, is a valuable addition. However, the central epsilon-regularity theorem (Theorem 4.3) is not proved in the paper but imported from Bellettini's De Giorgi iteration without verification of the required hypotheses; the proof of Theorem 1.8 also contains a gap concerning the claimed stationarity of the individual multi-valued graph varifolds. These gaps currently leave the main regularity theorem in a conditional state.

major comments (3)

[Section 4, Theorem 4.3] Theorem 4.3 is the keystone of the entire paper: it is used in Theorem 1.8 to obtain the Lipschitz multi-valued graphical decomposition, which in turn feeds the dimension-reduction argument for Theorem 1.3. Its proof is not contained in the manuscript; the text says one can follow Bellettini's argument after replacing constants, but the adaptation is not a mere bookkeeping change. Bellettini's iteration for a single-multiplicity hyperplane uses a Sobolev inequality on smooth minimal hypersurfaces; here the hypersurface M has H^{n-2}(Sing M)<∞, and the tilt function g has a product form with multiple zeros. The paper does not verify that the required Sobolev inequality holds with constants independent of t and of the branch set, nor that the level-set iteration works when the superlevel sets {g>t} are not smooth and may intersect the singular set. Proposition 4.2 supplies Caccioppoli ineq
[Section 4, proof of Theorem 1.8] Immediately after equation (1.6), the theorem asserts that each v(u_i) is stationary in B_{R/2}(0). The proof does not establish this. The multi-valued graph v(u_i) is obtained by grouping local embedded disks of the immersed hypersurface M according to which normal direction they are close to. Such a grouping cuts M along the level sets of the tilt function g, and the resulting pieces have boundary of positive (n-1)-measure in general; hence their varifolds are not automatically stationary. This matters later in the proof when the text identifies the degrees d_i by applying the Constancy Theorem to limits of the v(u_i). The paper should either prove stationarity of v(u_i) (if it is true in the present setting) or replace that argument with a consequence of the Lipschitz convergence and the density of the varifolds, which would still identify the limiting multiplicities.
[Section 6 and Appendix B] Theorem 1.5, which establishes sharpness in every dimension, depends crucially on Proposition B.1 and on the spectral lower bound Proposition 6.1. Proposition B.1 is stated as a consequence of Kapouleas-Wiygul's gluing estimates, but the proof is only a summary; the convergence in (B.3) and (B.6), as well as the eigenvalue convergence in Proposition 6.1, are used in a quantitative way. If any of these imported estimates are weaker than stated, the construction collapses. The authors should either provide more details for the derivation of Proposition B.1 from [KW22], or explicitly mark these statements as hypotheses inherited from the gluing construction. As it stands, the sharpness claim rests on an external result whose precise form is not independently verified here.

minor comments (4)

[Section 4, proof of Theorem 1.8] The phrase 'If either conclusion failed for arbitrarily small epsilon' is imprecise; the intended statement is 'for every epsilon>0 there exists a hypersurface satisfying the smallness assumption for which the conclusion fails.' The contradiction setup should be stated with a sequence epsilon_k -> 0.
[Section 4, Proposition 4.2] The integration by parts in (4.3) is performed on the superlevel set {g>t} without commenting on the fact that g is only Lipschitz (Proposition 4.1) and that {g=t} may be nonsmooth or meet Sing M. A short justification using the coarea formula or an approximation argument would make the proof rigorous.
[Appendix B, Lemma B.5] The passage from (B.20) to the boundary-trace statement v^{(a)}(0,cdot)=0 is terse. Since H^1 convergence is used, it would help to explicitly invoke the continuity of the trace operator and the weak lower semicontinuity of the L^2 norm on the boundary circle.
[Definition 1.1] The notation Sing M = (M \ M) ∩ B_2(0) uses both M as an immersion and M as the regular set; this is standard but could be clarified for the reader. In particular, the definition of Sing M should specify that M denotes the regular set of the immersion as a subset of the image.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular reduction found; self-citations are motivational and the central sheeting/regularity argument does not reduce to its assumptions.

full rationale

The main derivation chain is not circular. Theorem 1.7 constructs the tilt function G_p explicitly and proves the generalized Schoen inequality through internal algebraic estimates (Propositions 3.16–3.27); the self-citation [WZ26] is used only as motivation for the ansatz (3.3)–(3.8), not as the proof of that inequality. E_{C,R} is defined from the constructed tilt function, and Theorem 4.3's sup-bound is a genuine estimate rather than a restatement of the definition. The dependence on [Bel25] in Theorem 4.3 is an external, not self-citational, import: the paper says 'one can follow essentially the same argument as in [Bel25] to conclude', so the proof of the keystone epsilon-regularity is not written out in this manuscript. That is a proof-gap/correctness risk, especially because the current tilt function has a more complicated zero set, but it is not a circular step because it does not assume Theorem 1.3. Similarly, the sharpness construction in Theorem 1.5 relies on independent external results (Kapouleas–Wiygul gluing, Lawson's polar map) and on a spectral lower bound proved in Appendix B, not on the theorem being established. There are no fitted parameters later renamed as predictions, and no uniqueness theorem imported solely from the authors' prior work. Overall, the derivation is not equivalent to its inputs by construction; the only notable concern is the omitted proof of Theorem 4.3, which is an evidentiary gap rather than circularity.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The proof uses standard geometric measure theory background and external constructions. The genuinely new content—the generalized Schoen inequality and the branched sheeting theorem—is argued in the paper. The main borrowed load-bearing inputs are Bellettini's iteration and the Kapouleas–Wiygul gluing estimates.

assumptions (7)

domain assumption Stability inequality extends to Lipschitz test functions across H^{n−2}-finite singular sets via cutoffs
Used in Section 2.1 to justify testing with φ=(g−t)_+ψ in Proposition 4.2; the H^{n−2} finiteness supplies the cutoff construction.
standard math Allard compactness and Ahlfors regularity for stationary integral varifolds
Used in Section 5 to pass to varifold limits and to ensure nonzero varifold tangents.
standard math Federer dimension-reduction argument
Used at the end of Section 5 to convert pointwise exclusion of unpaired classical cones into the Hausdorff-dimension estimate.
standard math Structure theorem for stationary integral one-dimensional varifolds [AA76]
Used in Claim 2 of Theorem 1.3 to classify two-dimensional stationary cone links as sums of hyperplanes when no unpaired rays occur.
domain assumption Bellettini's De Giorgi iteration framework can be transplanted to the new tilt function
Theorem 4.3 is not reproved; the paper states it follows essentially by the same argument as [Bel25] after the Caccioppoli estimates of Proposition 4.2.
domain assumption Kapouleas–Wiygul gluing construction and its estimates
Underpins the sharp example in Theorem 1.5; the paper summarizes the construction and estimates in Appendix B rather than reproving the full gluing theorem.
standard math Lawson polar map properties and Simons cone-stability criterion
Used in Section 6 to convert the quadratic-form positivity (6.9) into stability of the cone over the polar map.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Regularity of branched stable minimal immersed hypersurfaces." pith.science (2026). https://pith.science/paper/QZVSXQM6

@misc{pith2026260729632,
  author       = {Pith},
  title        = {Pith review of: Regularity of branched stable minimal immersed hypersurfaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QZVSXQM6}},
  note         = {Machine review of arXiv:2607.29632}
}

abstract

We establish a sharp bound on the Hausdorff dimension of the non-branch singular set of branched stable minimal immersed hypersurfaces whose singular sets have locally finite $\mathcal H^{n-2}$-measure: the non-branch singular set is empty when $n=2$, discrete when $n=3$, and has Hausdorff dimension at most $n-3$ when $n\geq4$. We also construct a non-flat stable minimal cone in $\mathbb R^4$ arising from a branched minimal immersion whose vertex is a non-branch singularity. Taking products with Euclidean factors yields examples whose non-branch singular sets have Hausdorff dimension exactly $n-3$, showing that our regularity bound is sharp in every dimension $n\geq3$. The main ingredients in our proof are a generalized Schoen inequality and a corresponding branched sheeting theorem near stationary classical cones and unions of hyperplanes.

Figures

Figures reproduced from arXiv: 2607.29632 by the authors.

**Figure 1.** The map Ψ30,1,15 from a truncated Scherk cell to the corresponding cell in M30. The notation Kb in the figure emphasizes the dependence on the truncation parameter b; it is the cell denoted by K in the text. Finally, put the necks and their boundary as Nm = [ 2 i=1 [ 2m j=1 Km,i,j , Γm = [ 2 i=1 [ 2m j=1 Ψm,i,j (∂wingK). The closures of the four components of Mm \ Nm, called bulk regions, are denoted by Bm,1, . . .… view at source ↗

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

27 extracted references · 6 linked inside Pith

[1]

W. K. Allard and F. J. Almgren, Jr., The structure of stationary one dimensional varifolds with positive density, Invent. Math. 34 (1976), no. 2, 83--97. 425741

1976
[2]

Costante Bellettini, Extensions of S choen- S imon- Y au and S choen- S imon theorems via iteration \`a la D e G iorgi , Invent. Math. 240 (2025), no. 1, 1--34. 4871955

2025
[3]

Spencer Becker-Kahn, Paul Minter, and Neshan Wickramasekera, On the nature of stationary integral varifolds near multiplicity 2 planes, 2025, arXiv:2507.13148

arXiv 2025
[4]

Camillo De Lellis and Emanuele Nunzio Spadaro, Q -valued functions revisited , Mem. Amer. Math. Soc. 211 (2011), no. 991, vi+79. 2663735

2011
[5]

Nick Edelen and Paul Minter, Uniqueness of regular tangent cones for immersed stable hypersurfaces, Arch. Ration. Mech. Anal. 248 (2024), no. 6, Paper No. 121, 24. 4830551

2024
[6]

Han Hong, Haizhong Li, and Gaoming Wang, O n - S table M inimal H ypersurfaces in R ^ n+1 , 2024, arXiv.2407.03222

arXiv 2024
[7]

Brian Krummel, Paul Minter, and Neshan Wickramasekera, A B ranch S et S tratification for S tationary V arifolds with E psilon- R egularity , 2026, arXiv:2606.01511

arXiv 2026
[8]

Brian Krummel and Neshan Wickramasekera, Fine properties of branch point singularities: stationary two-valued graphs and stable minimal hypersurfaces near points of density <3 , 2021, arXiv:2111.12246

arXiv 2021

Show all 27 references

[9]

Nikolaos Kapouleas and David Wiygul, Minimal surfaces in the three-sphere by desingularizing intersecting C lifford tori , Math. Ann. 383 (2022), no. 1-2, 119--170. 4444117

2022
[10]

Blaine Lawson, Jr., Complete minimal surfaces in S 3 , Ann

H. Blaine Lawson, Jr., Complete minimal surfaces in S 3 , Ann. of Math. (2) 92 (1970), 335--374. 270280

1970
[11]

Paul Minter, The structure of stable codimension one integral varifolds near classical cones of density Q+1/2 , Calc. Var. Partial Differential Equations 63 (2024), no. 1, Paper No. 5, 28. 4668991

2024
[12]

, A C ampanato regularity theory for multi-valued functions with applications to minimal surface regularity theory , J. Funct. Anal. 289 (2025), no. 1, Paper No. 110908, 45. 4875500

2025
[13]

Differential Geom

, The structure of stable codimension one integral varifolds near classical cones of density 5/2 , J. Differential Geom. 129 (2025), no. 3, 725--863. 4871729

2025
[14]

Paul Minter and Neshan Wickramasekera, A structure theory for stable codimension 1 integral varifolds with applications to area minimising hypersurfaces mod \,p , J. Amer. Math. Soc. 37 (2024), no. 3, 861--927. 4736529

2024
[15]

Paul Minter and Zhengyi Xiao, An optimal regularity theory for immersed stable minimal hypersurfaces with small singular set, 2026, arXiv:2605.05041

2026 arXiv
[16]

Robert Osserman, A survey of minimal surfaces, second ed., Dover Publications, Inc., New York, 1986. 852409

1986
[17]

Richard Melvin Schoen, E XISTENCE AND REGULARITY THEOREMS FOR SOME GEOMETRIC VARIATIONAL PROBLEMS , ProQuest LLC, Ann Arbor, MI, 1977, Thesis (Ph.D.)--Stanford University. 2626915

1977
[18]

James Simons, Minimal varieties in R iemannian manifolds , Ann. of Math. (2) 88 (1968), 62--105. 233295

1968
[19]

Leon Simon, On a theorem of de G iorgi and S tampacchia , Math. Z. 155 (1977), no. 2, 199--204. 454857

1977
[20]

3, Australian National University, Centre for Mathematical Analysis, Canberra, 1983

, Lectures on geometric measure theory, Proceedings of the Centre for Mathematical Analysis, Australian National University, vol. 3, Australian National University, Centre for Mathematical Analysis, Canberra, 1983. 756417

1983
[21]

Pure Appl

Richard Schoen and Leon Simon, Regularity of stable minimal hypersurfaces, Comm. Pure Appl. Math. 34 (1981), no. 6, 741--797. 634285

1981
[22]

Schoen, L

R. Schoen, L. Simon, and S. T. Yau, Curvature estimates for minimal hypersurfaces, Acta Math. 134 (1975), no. 3-4, 275--288. 423263

1975
[23]

Differential Geom

Leon Simon and Neshan Wickramasekera, Stable branched minimal immersions with prescribed boundary, J. Differential Geom. 75 (2007), no. 1, 143--173. 2282727

2007
[24]

Pure Appl

, A frequency function and singular set bounds for branched minimal immersions, Comm. Pure Appl. Math. 69 (2016), no. 7, 1213--1258. 3503021

2016
[25]

Differential Geom

Neshan Wickramasekera, A regularity and compactness theory for immersed stable minimal hypersurfaces of multiplicity at most 2, J. Differential Geom. 80 (2008), no. 1, 79--173. 2434260

2008
[26]

, A general regularity theory for stable codimension 1 integral varifolds, Ann. of Math. (2) 179 (2014), no. 3, 843--1007. 3171756

2014
[27]

Gaoming Wang and Xuwen Zhang, Regularity of stable capillary minimal hypersurfaces, 2026, arXiv:2605.20964

2026 arXiv

Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.

[1] [1]

W. K. Allard and F. J. Almgren, Jr., The structure of stationary one dimensional varifolds with positive density, Invent. Math. 34 (1976), no. 2, 83--97. 425741

1976

[2] [2]

Costante Bellettini, Extensions of S choen- S imon- Y au and S choen- S imon theorems via iteration \`a la D e G iorgi , Invent. Math. 240 (2025), no. 1, 1--34. 4871955

2025

[3] [3]

Spencer Becker-Kahn, Paul Minter, and Neshan Wickramasekera, On the nature of stationary integral varifolds near multiplicity 2 planes, 2025, arXiv:2507.13148

arXiv 2025

[4] [4]

Camillo De Lellis and Emanuele Nunzio Spadaro, Q -valued functions revisited , Mem. Amer. Math. Soc. 211 (2011), no. 991, vi+79. 2663735

2011

[5] [5]

Nick Edelen and Paul Minter, Uniqueness of regular tangent cones for immersed stable hypersurfaces, Arch. Ration. Mech. Anal. 248 (2024), no. 6, Paper No. 121, 24. 4830551

2024

[6] [6]

Han Hong, Haizhong Li, and Gaoming Wang, O n - S table M inimal H ypersurfaces in R ^ n+1 , 2024, arXiv.2407.03222

arXiv 2024

[7] [7]

Brian Krummel, Paul Minter, and Neshan Wickramasekera, A B ranch S et S tratification for S tationary V arifolds with E psilon- R egularity , 2026, arXiv:2606.01511

arXiv 2026

[8] [8]

Brian Krummel and Neshan Wickramasekera, Fine properties of branch point singularities: stationary two-valued graphs and stable minimal hypersurfaces near points of density <3 , 2021, arXiv:2111.12246

arXiv 2021