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Math-2446 · mathematics · Pde

Almost Everywhere Regularity of Parametrized Stationary Varifolds

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Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
For a general parametrized integer stationary $n$-varifold, does the singular set have zero $n$-dimensional Hausdorff measure without additional hypotheses?

Formulation references: [S1], [S2].

Short English statement

For a general parametrized integer stationary $n$-varifold, does the singular set have zero $n$-dimensional Hausdorff measure without additional hypotheses?

Sources

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

Math-2446-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2446

Codimension-one varifold singular bound: The manuscript assumes stationarity under variations of the ambient space for an integral varifold. The word parametrized in this question has not been established here to impose exactly that stationarity notion, so transferring the general assertion requires an identification of the two conventions.

Almost-everywhere varifold regularity: The manuscript assumes stationarity under variations of the ambient space for an integral varifold. The word parametrized in this question has not been established here to impose exactly that stationarity notion, so transferring the general assertion requires an identification of the two conventions.

This page records partial support. Complete resolution of the catalogue question is not established by this record.

Almost Everywhere Regularity of Parametrized Stationary Varifolds

OpenAI's claim

Codimension-one varifold singular bound: For a stationary integral varifold of dimension $m$, the refinement claims a singular-set Hausdorff bound of $m-1$; the Euclidean ambient dimension can be any larger dimension, and the assertion begins at $m=1$.

Almost-everywhere varifold regularity: For Euclidean stationary integral varifolds of dimension $m$, the initial paper claims that singular points carry no $m$-dimensional Hausdorff measure; off this exceptional set there are smooth minimal neighborhoods with one constant integer multiplicity; a separate argument handles round spheres.

Mathematical statement of the claimed result

Mathematical claim 1.
Let $V$ be a stationary integral $m$-varifold in open $U\subset\mathbb R^{m+n}$, with $m,n\ge1$. Its weight has the form $\mu=\theta\mathcal H^m|_E$ with integer multiplicity, and stationarity requires $\int\operatorname{tr}(P_{T_xV}DX)\,d\mu=0$ for every compactly supported $C^1$ ambient vector field. Define singular points as support points without a neighborhood equal to one smooth embedded minimal submanifold with constant positive integer multiplicity. The claimed bound is \[\dim_H\operatorname{Sing}V\le m-1.\] Stability, minimization and a codimension-one assumption are not required.
Mathematical claim 2.
For arbitrary integers $m,n\ge1$, take an integral varifold $V$ of dimension $m$ on some open $U\subset\mathbb R^{m+n}$. If its first variation vanishes, it is claimed to satisfy \[\mathcal H^m(\operatorname{Sing}V)=0.\] Regularity here requires the local support to be a single smooth embedded minimal manifold with constant integer multiplicity. No stability, minimizing property, orientability, density upper bound or codimension restriction is imposed.

Outline of the argument

Codimension-one varifold singular bound: Smooth reference graphs are fitted at prescribed flat scales, with controlled frequency intervals covering transitions between scales. Centered heights are normalized so that their second moments remain nonzero while their first moments vanish. The limiting height and small mean-curvature errors provide the dimension-reduction estimate needed to sharpen the earlier almost-everywhere result.

Almost-everywhere varifold regularity: Integer multiplicity, height estimates, and heat-contact comparisons are combined in a signed-excess induction. Packing of witness scales closes the estimates when sheets merge. A center chosen from minimal fits and multiplicity defects gives inward doubling, forcing the residual height to vanish and the exceptional support to have zero relevant Hausdorff measure.

Formal verification

Codimension-one varifold singular bound: The pinned catalogue lists no Lean formalization for this result.

Almost-everywhere varifold regularity: The pinned catalogue lists no Lean formalization for this result.

This site has not run the Lean development. These links describe OpenAI’s selected formal statements and should be read with their stated scope.

Status caveat

OpenAI reports that an internal model produced the manuscripts. Their arguments have not been independently verified as proofs by this site. Any portion outside the documented formal scope may contain mistakes, and the catalogue label reports the announced mathematical claim.

Sources

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