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arxiv: 2606.01511 · v1 · pith:5QHKLH3Dnew · submitted 2026-06-01 · 🧮 math.DG · math.AP

A Branch Set Stratification for Stationary Varifolds with Epsilon-Regularity

Brian Krummel , Paul Minter , Neshan Wickramasekera This is my paper

Pith reviewed 2026-06-28 13:11 UTC · model grok-4.3

classification 🧮 math.DG math.AP
keywords stationary varifoldsbranch pointsHausdorff dimensionepsilon-regularityintegral varifoldsmultiplicitystratificationfrequency function
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The pith

Stationary integral n-varifolds in classes closed under limits and satisfying epsilon-regularity near planes of multiplicity at most Q have their density-Q branch points confined to a set of Hausdorff dimension at most n-2.

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

The paper establishes that any class of stationary integral n-varifolds closed under weak limits, homotheties, rotations and disjoint decompositions, and obeying an epsilon-regularity condition near planes of multiplicity at most Q, must have its branch points of density at most Q forming a set whose Hausdorff dimension is at most n-2. This bound applies in particular to graphs of two-valued Lipschitz functions and to stable codimension-one varifolds that have no classical singularities of density less than Q. A reader would care because branch points mark locations where the varifold fails to be a smooth multiple cover of a manifold, and a dimension bound limits how much they can contribute to the overall singular set. The argument proceeds via the planar frequency function, avoiding the full Almgren center manifold construction except in one canonical geometric case.

Core claim

Suppose V is a class of stationary integral n-varifolds in the ball B^{n+k}_2(0) that is closed under weak limits, homotheties, rotations and disjoint decomposition, and suppose V satisfies the epsilon-regularity property near planes of multiplicity at most Q. Then for every varifold in V the set of branch points with density at most Q has Hausdorff dimension at most n-2. Direct consequences are the same dimension bound for the density-2 branch set of any stationary 2-valued Lipschitz graph and for the density-Q branch set of any stable codimension-one stationary integral varifold with no classical singularities of density less than Q.

What carries the argument

The epsilon-regularity property near planes of multiplicity at most Q, which requires that varifolds sufficiently close to such a plane in the unit cylinder are represented in the half-cylinder by graphs of Lipschitz multi-valued functions obeying uniform C^{1,alpha} estimates.

If this is right

  • The density-2 branch set of any stationary integral varifold that is the graph of a two-valued Lipschitz function has Hausdorff dimension at most n-2.
  • The density-Q branch set of any stable codimension-one stationary integral varifold with no classical singularities of density less than Q has Hausdorff dimension at most n-2.
  • The planar frequency function suffices to obtain the dimension bound on branch points except in one geometrically canonical case where the center manifold has simplifying properties.

Where Pith is reading between the lines

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

  • The same dimension control would hold for any larger class of varifolds once the epsilon-regularity property is verified for that class.
  • Combining the bound with existing removal theorems for classical singularities could produce sharper statements about the entire singular set in the stable codimension-one case.
  • The frequency-function approach might adapt directly to related stratification questions for varifolds that are stationary with respect to other functionals.

Load-bearing premise

The class of varifolds must satisfy the epsilon-regularity condition that forces any member epsilon-close to a plane of multiplicity at most Q to be a multi-valued Lipschitz graph with quantitative C^{1,alpha} estimates.

What would settle it

Exhibit a varifold belonging to one of the classes V (closed under the listed operations and satisfying epsilon-regularity) whose set of branch points with density at most Q has Hausdorff dimension strictly larger than n-2.

read the original abstract

Suppose $\mathcal{V}$ is a class of stationary integral $n$-varifolds in $B^{n+k}_2(0)\subset\mathbb{R}^{n+k}$ which is closed under weak limits, homotheties, rotations, and disjoint decomposition, and suppose that $\mathcal {V}$ satisfies an $\epsilon$-regularity property near planes of (integer) multiplicity $\leq Q\in \{2,3,\dotsc\}$. This last condition, more precisely, requires that there be a constant $\epsilon = \epsilon({\mathcal V}, Q) \in (0, 1)$ such that if $V\in \mathcal{V}$ is, in the unit cylinder ${\mathbb R}^{k} \times B_{1}^{n}(0)$, $\epsilon$-close as varifolds to the plane $\{0\} \times {\mathbb R}^{n}$ taken with multiplicity $\leq Q$ then, in the half-cylinder ${\mathbb R}^{k} \times B_{1/2}^{n}(0)$, $V$ is represented by the graph of a Lipschitz multi-valued function over $B_{1/2}^{n}(0)$ with uniform quantitative estimates of a $C^{1,\alpha}$ nature. For any varifold in such a class $\mathcal{V}$, we prove that the set of branch points with density $\leq Q$ has Hausdorff dimension $\leq n-2$. By choosing suitable $\mathcal{V}$, a direct consequence of this result and the recently established regularity theorems of the second and third authors (one of which being joint with Becker-Kahn) is that if $V$ is a stationary integral $n$-varifold which is either: (a) represented by the graph of a $2$-valued Lipschitz function; or (b) codimension one, stable, and with no classical singularities of density $<Q$, then the Hausdorff dimension of the density $Q$ branch set ($Q=2$ in (a)) is at most $n-2$. Our proof utilises the planar frequency function introduced by the first and third authors in their work on area minimising currents, and thus does not require the Almgren center manifold for the analysis of branch points except in a single, geometrically canonical case where the center manifold satisfies additional simplifying properties.

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

0 major / 3 minor

Summary. The paper proves that if V is a class of stationary integral n-varifolds in the 2-ball of R^{n+k} that is closed under weak limits, homotheties, rotations and disjoint decomposition, and that satisfies an epsilon-regularity hypothesis near planes of integer multiplicity at most Q (specifically, epsilon-close varifolds in the unit cylinder are represented in the half-cylinder by graphs of Lipschitz multi-valued functions with uniform C^{1,alpha} estimates), then the branch points of density at most Q have Hausdorff dimension at most n-2. Direct consequences are stated for the branch sets of 2-valued Lipschitz graphs and of codimension-one stable varifolds without classical singularities of density less than Q. The argument adapts the planar frequency function of the first and third authors, restricting the center-manifold construction to a single geometrically canonical case.

Significance. If the result holds, it supplies a dimension bound on branch sets for stationary varifolds under a natural epsilon-regularity closure assumption, thereby extending the frequency-function techniques previously developed for area-minimizing currents to a broader setting. The conditional nature of the statement (tied directly to the epsilon-regularity axiom) and the limited use of the center manifold are strengths that keep the argument self-contained relative to the stated hypotheses. The applications to multi-valued graphs and stable hypersurfaces furnish concrete new regularity statements once the epsilon-regularity is verified in those classes.

minor comments (3)
  1. [Abstract] Abstract, paragraph 2: the phrase 'represented by the graph of a 2-valued Lipschitz function' should be cross-referenced to the precise definition of the class V used in the application; the current wording leaves open whether the epsilon-regularity is assumed or derived for that class.
  2. [Section 3 (assumed from context)] The manuscript invokes the planar frequency function from prior work but does not include a self-contained statement of the monotonicity formula or the precise form of the frequency used in the varifold setting; adding a short appendix or subsection recalling the relevant identities would improve readability.
  3. [Abstract / Introduction] The epsilon-regularity hypothesis is stated in terms of 'epsilon-close as varifolds' and 'uniform quantitative estimates of a C^{1,alpha} nature'; a brief remark clarifying the precise varifold distance (e.g., F-metric or mass distance in the cylinder) and the dependence of alpha on Q would remove ambiguity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, assessment of significance, and recommendation of minor revision. No specific major comments appear in the report, so we have no points requiring response or revision at this stage.

Circularity Check

1 steps flagged

Minor self-citation of frequency function, not load-bearing on central claim

specific steps
  1. self citation load bearing [Abstract]
    "Our proof utilises the planar frequency function introduced by the first and third authors in their work on area minimising currents, and thus does not require the Almgren center manifold for the analysis of branch points except in a single, geometrically canonical case where the center manifold satisfies additional simplifying properties."

    This cites prior work by two current authors for the frequency function used in the proof; however the citation is not load-bearing because the present argument is explicitly conditional on the new epsilon-regularity hypothesis for V and does not derive the Hausdorff dimension bound from the citation alone.

full rationale

The paper adapts the planar frequency function from prior work by the first and third authors but applies it conditionally on the epsilon-regularity and closure axioms of class V. The stratification result does not reduce any prediction to a fitted input or self-defined quantity, nor does it rely on a uniqueness theorem imported from the same authors. The self-citation supports an established analytic tool rather than carrying the dimension bound itself, leaving the derivation self-contained against the stated assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the closure properties of the class V and the epsilon-regularity hypothesis; no free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption The class V is closed under weak limits, homotheties, rotations, and disjoint decomposition.
    Stated explicitly in the abstract as part of the setup for the class of varifolds.
  • domain assumption V satisfies an epsilon-regularity property near planes of multiplicity ≤ Q, with the precise graphical representation and C^{1,alpha} estimates in the half-cylinder.
    This is the key hypothesis invoked to conclude the dimension bound on branch points.

pith-pipeline@v0.9.1-grok · 5970 in / 1445 out tokens · 27941 ms · 2026-06-28T13:11:32.296665+00:00 · methodology

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Reference graph

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