Perpendicularity and Locality for Codimension-One Varifolds with Bounded Anisotropic Mean Curvature

Mario Santilli; S{\l}awomir Kolasi\'nski

arxiv: 2603.21983 · v2 · submitted 2026-03-23 · 🧮 math.AP · math.DG· math.OC

Perpendicularity and Locality for Codimension-One Varifolds with Bounded Anisotropic Mean Curvature

S{\l}awomir Kolasi\'nski , Mario Santilli This is my paper

Pith reviewed 2026-05-15 00:49 UTC · model grok-4.3

classification 🧮 math.AP math.DGmath.OC

keywords varifoldsanisotropic curvaturemean curvaturerectifiabilitynormal bundlecodimension oneanisotropic integrandsfirst variation

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The pith

The mean F-curvature of these varifolds lies normal to the support surface almost everywhere and matches its rectifiable approximation.

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

The paper establishes that when an n-dimensional varifold in R^{n+1} has bounded anisotropic mean F-curvature coming from a uniformly convex C^3 integrand, its curvature vector points strictly in the normal direction at almost every point of the C^2-rectifiable part of its support. This property also ensures that the varifold curvature coincides with the approximate mean curvature computed from the rectifiable structure itself. The result extends classical Euclidean theorems on varifolds to the anisotropic setting, where the underlying norm is no longer the Euclidean one. A sympathetic reader would care because the normal condition is the key geometric constraint that makes curvature-driven evolution and minimality questions well-posed for these generalized surfaces.

Core claim

We prove that h_F(V,a) belongs to Nor(M,a) for H^n-almost every a in M and that h_F(V,·) agrees with the approximate mean F-curvature obtained from the C^2-rectifiable covering of M.

What carries the argument

The anisotropic mean curvature h_F(V,·) of the varifold, shown to lie in the normal bundle of the C^2-rectifiable support M.

Load-bearing premise

F comes from a uniformly convex C^3-norm and the varifold has absolutely continuous support measure with bounded L^infty mean F-curvature.

What would settle it

A concrete varifold satisfying the bounded-curvature hypotheses whose mean F-curvature vector has a nonzero tangential component on a positive H^n-measure set inside the C^2-rectifiable part.

read the original abstract

Suppose $ F $ is an integrand associated with a uniformly convex $ \mathscr{C}^{3} $-norm, and $ V $ is a $ n $-dimensional varifold in an open subset of $ \mathbf{R}^{n+1} $ such that $ \mathscr{H}^n \llcorner \operatorname{spt} \| V \| $ is absolutely continuous with respect to $ \| V \| $ and the mean $ F $-curvature $ \mathbf{h}_{F}(V, \cdot) $ is bounded in $\mathbf{L}^\infty $. In our previous result arXiv:2507.18357 we prove that $ \operatorname{spt} \| V \| $ is $ \mathscr{C}^{2} $-rectifiable and the $ \mathscr{C}^{1} $-regular part $ M $ of $ \operatorname{spt} \| V \| $ coincides $ \mathscr{H}^n $ almost everywhere with the unit-density stratum of $ V $. In this paper we prove that $ \mathbf{h}_{F}(V,a) \in \operatorname{Nor}(M,a) $ for $ \mathscr{H}^n $ a.e.\ $ a \in M $ and that $ \mathbf{h}_{F}(V, \cdot) $ agrees with the approximate mean $ F $-curvature coming from the $ \mathscr{C}^{2} $-rectifiable covering of $ M $. These results provide anisotropic extensions of well known theorems in the Euclidean setting by Brakke, Sch\"atzle and Ambrosio-Masnou.

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.

Desk Editor's Note private letter to a colleague

This paper cleanly extends the Brakke-Schätzle perpendicularity and locality theorems to anisotropic varifolds with bounded mean F-curvature, using the authors' prior rectifiability result.

read the letter

This paper extends the classical perpendicularity and locality results for varifolds to the anisotropic setting. It proves that the mean F-curvature vector lies in the normal space to the C^2-rectifiable set almost everywhere and agrees with the approximate mean F-curvature from the rectifiable covering. The work builds directly on the authors' earlier rectifiability theorem under the assumptions of a uniformly convex C^3-norm for F, absolute continuity of Hausdorff measure with respect to the varifold measure, and an L^infty bound on the curvature. The first-variation formula and density arguments carry over in a natural way once rectifiability is in hand. The paper does well by stating the claims precisely and positioning the results as straightforward extensions rather than claiming broader novelty. The setup mirrors the Euclidean case closely enough that the adaptation looks feasible. The main soft spot is the heavy dependence on the companion paper for C^2-rectifiability and unit density; if that result holds, this one should follow without new gaps, but a referee would still want explicit verification that anisotropy does not introduce fresh technical issues in the estimates. The abstract gives no proof sketches, so the full manuscript needs checking for any post-hoc choices or hidden regularity assumptions. This is for specialists in geometric measure theory working on anisotropic variational problems. A reader already familiar with the Euclidean theorems will get value from seeing the adaptation. It deserves a serious referee because the claims are focused, the hypotheses are standard, and the extension fills a clear gap in the literature.

Referee Report

0 major / 3 minor

Summary. The manuscript proves that for an n-dimensional varifold V in an open subset of R^{n+1} whose support satisfies H^n ≪ ||V|| and whose anisotropic mean curvature h_F(V,·) is bounded in L^∞, with F arising from a uniformly convex C^3-norm, the vector h_F(V,a) lies in the normal space Nor(M,a) for H^n-a.e. a in the C^2-rectifiable part M of spt||V||, and moreover h_F coincides H^n-a.e. with the approximate mean F-curvature obtained from the C^2-rectifiable covering of M. The argument relies on the authors' prior C^2-rectifiability theorem (arXiv:2507.18357) together with the first-variation formula for the anisotropic functional and standard density arguments.

Significance. If the claims hold, the work supplies the natural anisotropic counterparts of the classical perpendicularity and locality results of Brakke, Schätzle, and Ambrosio-Masnou. These identifications are load-bearing for any subsequent analysis of anisotropic curvature flows or stationary varifolds with bounded mean curvature, and the paper correctly credits the structural input from the companion rectifiability theorem.

minor comments (3)

[Introduction] §1, line 3: the phrase 'the C^1-regular part M of spt||V|| coincides H^n-a.e. with the unit-density stratum' should be restated with an explicit reference to the precise statement in the companion paper arXiv:2507.18357 to avoid any ambiguity about the measure-theoretic identification.
[Main results] The notation for the approximate mean F-curvature (introduced after the rectifiable covering) is used without a displayed definition; adding a short displayed equation would improve readability for readers unfamiliar with the Euclidean precursors.
[Theorem statement] Theorem 1.1 (or equivalent): the statement that h_F agrees with the approximate curvature should include a parenthetical remark on the precise notion of 'approximate' (e.g., via blow-up or density ratios) to match the level of detail in the cited Euclidean references.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of the manuscript, including the recognition that the results supply natural anisotropic counterparts to the classical perpendicularity and locality theorems. We note the recommendation for minor revision.

Circularity Check

0 steps flagged

Minor self-citation for structural input; new claims independent

full rationale

The paper invokes its own prior arXiv:2507.18357 solely to obtain C^2-rectifiability and unit-density strata of the support, then applies the first-variation formula for the anisotropic functional together with standard density and approximate-tangent arguments to conclude that h_F lies in the normal space and agrees with the classical curvature. These steps are direct extensions of Euclidean GMT techniques (Brakke, Schätzle) and do not reduce the new statements to algebraic identities or fitted quantities defined in terms of themselves. The self-citation supplies an external structural hypothesis rather than a load-bearing loop internal to the present derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper operates entirely within the established framework of varifold theory and rectifiability; no new free parameters, ad-hoc constants, or postulated entities are introduced.

axioms (2)

standard math Standard varifold theory and rectifiability results from Allard, Brakke, and related literature
Invoked to pass from bounded curvature to C^2-rectifiability of the support.
domain assumption Uniform convexity and C^3 regularity of the norm defining the integrand F
Stated explicitly as the setting for the anisotropic curvature.

pith-pipeline@v0.9.0 · 5594 in / 1365 out tokens · 27509 ms · 2026-05-15T00:49:35.222578+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

h_F(V,a) ∈ Nor(M,a) for H^n a.e. a ∈ M and that h_F(V,·) agrees with the approximate mean F-curvature coming from the C^2-rectifiable covering of M
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

uniformly convex C^3-norm … bounded L^∞ mean F-curvature

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