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arxiv: 2509.21508 · v2 · submitted 2025-09-25 · 🧮 math.DG · math.AP

Stationary and stable varifolds with singularities

Camillo De Lellis , Jonas Hirsch , Luca Spolaor This is my paper

Pith reviewed 2026-05-18 13:42 UTC · model grok-4.3

classification 🧮 math.DG math.AP
keywords minimal immersionsvarifoldssingularitiescatenoidal necksfloating disksC^{1,α} metricsstationary varifoldsstable varifolds
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The pith

Minimal m-dimensional immersions in R^{m+1} with C^{1,α} metrics can have sequences of catenoidal necks or floating disks converging to isolated multiplicity-2 flat singular points.

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

This paper constructs examples of minimal immersions of m-dimensional manifolds into Euclidean space equipped with a metric of limited smoothness. The immersions are stationary and stable but feature a sequence of catenoidal necks or floating disks that converge to a single isolated singular point. At this point the surface is flat and has multiplicity two. Such constructions matter because they illustrate how singularities can form in minimal surfaces even when the ambient space has only C^{1,α} regularity for α less than one.

Core claim

The authors construct minimal m-dimensional immersions in R^{m+1} equipped with a C^{1,α} metric for α in [0,1) that include a sequence of catenoidal necks or floating disks converging to an isolated, multiplicity 2, singular flat point. These immersions remain minimal, stationary, and stable despite the presence of the singularity.

What carries the argument

The sequence of catenoidal necks or floating disks that converges to an isolated multiplicity-2 flat singular point, allowing the preservation of minimality and stability in the C^{1,α} metric.

If this is right

  • These examples show that stable minimal varifolds can possess isolated flat singularities under low-regularity metrics.
  • The convergence of necks or disks maintains the stationarity and stability properties.
  • Such singularities are possible for hypersurfaces in any dimension m.
  • The construction works for metrics of regularity C^{1,α} with α in [0,1).

Where Pith is reading between the lines

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

  • This suggests potential for constructing more complex singular minimal surfaces in metrics with even less regularity.
  • Similar techniques might apply to other types of singularities like branched points or higher multiplicity.
  • These constructions could serve as test cases for numerical simulations of minimal surface evolution or regularity questions.

Load-bearing premise

The construction assumes that sequences of catenoidal necks or floating disks can be arranged to converge to the isolated multiplicity-2 flat singular point while preserving minimality, stationarity, and stability under the given C^{1,α} metric regularity.

What would settle it

A direct computation or numerical approximation of the constructed immersion near the limit point could check whether the mass density is exactly 2 and whether the tangent cone is a flat plane with multiplicity 2, or if the stability inequality holds in the limit.

read the original abstract

We construct minimal $m$-dimensional immersions in $\R^{m+1}$, equipped with a $C^{1, \alpha}$ metric, $\alpha\in [0,1)$, with a sequence of \emph{catenoidal necks} or \emph{floating disks} converging to an isolated, multiplicity $2$, singular flat point.

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

1 major / 1 minor

Summary. The manuscript constructs minimal m-dimensional immersions into R^{m+1} equipped with a C^{1,α} metric (α ∈ [0,1)) that contain sequences of catenoidal necks or floating disks converging to an isolated multiplicity-2 flat singular point; the resulting limit objects are asserted to be stationary and stable varifolds with singularities.

Significance. If the stability of the limit varifold is rigorously established, the construction would supply explicit examples of stable singular minimal varifolds in ambient spaces whose metrics have only Lipschitz regularity. This lies below the C^2 threshold usual for classical minimal hypersurface theory and could serve as a test case for the extent to which stationarity and stability persist under weak convergence in low-regularity metrics.

major comments (1)
  1. [Abstract] Abstract: the central claim that the limit varifold remains stable requires that the non-negativity of the second variation pass to the limit. For α < 1 the metric is merely Lipschitz, so the Jacobi operator is not classically elliptic; the manuscript must supply quantitative estimates on the rate at which the necks (or disks) shrink in order to justify integration-by-parts identities or a modified stability inequality in the limit.
minor comments (1)
  1. [Abstract] The abstract states the existence of the construction but supplies no outline of the approximation scheme or error controls; a brief sketch in the introduction would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the insightful comment on the stability of the limiting varifold. The concern about passing the second variation to the limit under merely Lipschitz metrics is well-taken, and we address it directly below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that the limit varifold remains stable requires that the non-negativity of the second variation pass to the limit. For α < 1 the metric is merely Lipschitz, so the Jacobi operator is not classically elliptic; the manuscript must supply quantitative estimates on the rate at which the necks (or disks) shrink in order to justify integration-by-parts identities or a modified stability inequality in the limit.

    Authors: We agree that additional justification is required to pass stability to the limit when the ambient metric is only C^{1,α} with α < 1. In the revised manuscript we will insert a new subsection (Section 4.3) that derives explicit quantitative bounds on the neck (or disk) radii in terms of their distance to the singular point. These bounds are obtained from the explicit construction of the approximating minimal immersions and are strong enough to control the error terms arising from the non-smooth metric in the second-variation formula. Using these estimates we establish a modified stability inequality for the approximating surfaces that passes to the limit varifold, thereby justifying the non-negativity of the second variation for the stationary limit object. The abstract will be updated to reflect this added justification. revision: yes

Circularity Check

0 steps flagged

Direct construction of varifolds with no self-referential reduction

full rationale

The paper presents an explicit construction of minimal immersions with catenoidal necks or floating disks converging to a multiplicity-2 singular point under a C^{1,α} metric. No equations, fitted parameters, or derivations appear in the provided abstract or description that reduce a claimed output to an input by definition or self-citation. The result is framed as a direct existence statement rather than a prediction derived from prior fitted quantities or uniqueness theorems internal to the authors' work. Stationarity and stability are asserted to hold in the limit by the construction itself, without evidence of circular redefinition or load-bearing self-citation chains. This is a standard non-circular construction result in geometric analysis.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract alone supplies no explicit free parameters, axioms, or invented entities; the result is framed as an existence construction whose internal assumptions are not detailed.

pith-pipeline@v0.9.0 · 5571 in / 1062 out tokens · 51117 ms · 2026-05-18T13:42:50.949438+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages

  1. [1]

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

    work page 1972

  2. [2]

    On the nature of stationary integral varifolds near multiplicity 2 planes, 2025

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

    work page 2025

  3. [3]

    Extensions of Schoen-Simon-Yau and Schoen-Simon theorems via iteration ` a la De Giorgi.Invent

    Costante Bellettini. Extensions of Schoen-Simon-Yau and Schoen-Simon theorems via iteration ` a la De Giorgi.Invent. Math., 240(1):1–34, 2025

    work page 2025

  4. [4]

    Brakke.The motion of a surface by its mean curvature, volume 20 ofMathematical Notes

    Kenneth A. Brakke.The motion of a surface by its mean curvature, volume 20 ofMathematical Notes. Princeton University Press, Princeton, NJ, 1978

    work page 1978

  5. [5]

    Remarks and Conjectures on Stationary Varifolds

    Camillo Brena, Stefano Decio, and Camillo De Lellis. Remarks and Conjectures on Stationary Varifolds. Preprint, arXiv: 2503.00651, 2025

    work page arXiv 2025

  6. [6]

    Camillo Brena, Camillo De Lellis, and Federico Franceschini.c ∞ rectifiability of stationary varifolds, 2025

    work page 2025

  7. [7]

    Minimal surfaces with isolated singularities.Manuscr

    Luis Caffarelli, Robert Hardt, and Leon Simon. Minimal surfaces with isolated singularities.Manuscr. Math., 48:1–18, 1984

    work page 1984

  8. [8]

    Allard’s interior regularity theorem: an invitation to stationary varifolds.Proceedings of CMSA Harvard

    Camillo De Lellis. Allard’s interior regularity theorem: an invitation to stationary varifolds.Proceedings of CMSA Harvard. Nonlinear analysis in geometry and applied mathematics. Part 2, pages 23–49, 2018

    work page 2018

  9. [9]

    Nonclassical minimizing surfaces with smooth boundary.J

    Camillo De Lellis, Guido De Philippis, and Jonas Hirsch. Nonclassical minimizing surfaces with smooth boundary.J. Differential Geom., 122:205 – 222, 2022

    work page 2022

  10. [10]

    Area minimizing hypersurfaces modulop: a geometric free-boundary problem.accepted in Journal of Functional Analysis, 2025

    Camillo De Lellis, Jonas Hirsch, Andrea Marchese, Luca Spolaor, and Salvatore Stuvard. Area minimizing hypersurfaces modulop: a geometric free-boundary problem.accepted in Journal of Functional Analysis, 2025

    work page 2025

  11. [11]

    A short proof of Allard’s and Brakke’s regularity theorems.Int

    Guido De Philippis, Carlo Gasparetto, and Felix Schulze. A short proof of Allard’s and Brakke’s regularity theorems.Int. Math. Res. Not. IMRN, (9):7594–7613, 2024

    work page 2024

  12. [12]

    Dimension of the singular set for 2-valued stationary lipschitz graphs, 2023

    Jonas Hirsch and Luca Spolaor. Dimension of the singular set for 2-valued stationary lipschitz graphs, 2023

    work page 2023

  13. [13]

    Measure 0 of the singular set for 2-valued stationary hypercurrents, 2025

    Jonas Hirsch and Luca Spolaor. Measure 0 of the singular set for 2-valued stationary hypercurrents, 2025

    work page 2025

  14. [14]

    Second order rectifiability of integral varifolds of locally bounded first variation.J

    Ulrich Menne. Second order rectifiability of integral varifolds of locally bounded first variation.J. Geom. Anal., 23(2):709– 763, 2013

    work page 2013

  15. [15]

    O. Savin. Viscosity solutions and the minimal surface system. InNonlinear analysis in geometry and applied mathematics. Part 2, volume 2 ofHarv. Univ. Cent. Math. Sci. Appl. Ser. Math., pages 135–145. Int. Press, Somerville, MA, 2018

    work page 2018

  16. [16]

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

    Leon Simon.Lectures on geometric measure theory, volume 3 ofProceedings of the Centre for Mathematical Analysis, Australian National University. Australian National University, Centre for Mathematical Analysis, Canberra, 1983

    work page 1983

  17. [17]

    A frequency function and singular set bounds for branched minimal immersions

    Leon Simon and Neshan Wickramasekera. A frequency function and singular set bounds for branched minimal immersions. Comm. Pure Appl. Math., 69(7):1213–1258, 2016. STATIONARY AND STABLE VARIFOLDS WITH SINGULARITIES 13 Institute for Advanced Study, 1 Einstein Drive, Princeton NJ 08540, USA Email address:camillo.delellis@math.ias.edu Mathematisches Institut,...

    work page 2016