gpt 6 astra ultra attempt on Math-418

Measure-zero singular set of stationary integral varifolds

Definitions and mathematical statement

An integral $m$-varifold in $U\subset{\mathbb{R}}^{m+n}$ is a Radon measure on position--tangent-plane pairs represented by a countably $m$-rectifiable set with locally integrable positive integer multiplicity. It is stationary when its first variation vanishes: \[ \int\operatorname{div}_P X(x){\,\mathrm{d}} V(x,P)=0 \quad\text{for every }X\in C_c^1(U;{\mathbb{R}}^{m+n}). \] A regular point of $\operatorname{spt}V\cap U$ has a neighborhood on which $V$ is an integer multiple of a smooth embedded minimal $m$-submanifold. Let $\operatorname{Sing}(V)$ be the remaining support points. The conjecture is $\mathcal H^m(\operatorname{Sing}(V))=0$ for all positive $m,n$. Stationarity alone is assumed: area minimization, stability, and multiplicity one are not extra hypotheses. Rectifiability almost everywhere is weaker than having these open smooth neighborhoods.

Formulation and concept references: [S1].

Short English statement

Does every stationary integral varifold have a singular set of zero measure in its own dimension, without any stability or minimizing assumption?

Research attempt

Scope, attribution, and checked context.

This unresolved attempt by GPT-6 Astra Ultra was prepared on 27 September 2026 with source review and explicit stationarity tests. Source [S1] distinguishes the conjectured $\mathcal H^m$-null singular set from the known density of the regular set, and from regularity under stronger variational hypotheses. Those distinctions are retained. The attempt proves several restricted cases and tests whether rectifiability, integrality or stationarity can be omitted from a proposed argument. No area-minimizing or stability assumption is added.

What density one already gives.

We use the classical Allard regularity theorem in the following standard stationary integral setting: if a blow-up at a point is a multiplicity-one plane, sufficiently small mass and tilt excess at small scales give a smooth embedded minimal graph near that point. The $C^{1,\alpha}$ conclusion is upgraded to smoothness by the minimal surface equation. This is an established input recorded in [S1], not an elementary theorem proved here.

At almost every point of an integral varifold, rectifiability gives a tangent plane with an integer multiplicity $Q$. At points with $Q=1$, that blow-up meets the multiplicity-one hypothesis. Consequently the singular points of density one form a $\|V\|$-null set. The unresolved part concerns points of higher multiplicity; dividing the tangent plane by $Q$ does not divide the nearby varifold into $Q$ smooth sheets.

For stationary integral varifolds, monotonicity gives a lower mass-density bound at every support point, by approximating it with rectifiable points of integer density and applying the monotonicity inequality. The usual covering argument then makes $\mathcal H^m$ on the support controlled locally by $\|V\|$. In particular replacing a mass-null exceptional set by an $\mathcal H^m$-null one here does not discard an additional large invisible part of the support.

A useful constant-multiplicity subcase.

Suppose $p$ has tangent varifold $Q|P|$ and there is a neighborhood $U_p$ in which the rectifiable multiplicity of $V$ equals the same integer $Q$ almost everywhere. Then $W=V/Q$ is integral and stationary on $U_p$ and has multiplicity-one tangent $|P|$ at $p$. Allard regularity applies to $W$ and proves that $V$ is $Q$ times a smooth minimal graph near $p$. Thus $p$ is regular in precisely the sense of the catalogue.

The hypothesis is local equality of the multiplicity, not just the statement that its approximate limit at $p$ is $Q$. A measurable integer-valued function is approximately continuous almost everywhere, but need not be constant on any open neighborhood there. Small sets of other multiplicities may persist at every scale. Replacing approximate constancy by open local constancy would silently assume the main regularity issue.

Stationarity forbids an easy multiplicity counterexample.

Let the support be one affine plane $P$, and write $V=\theta\,\mathcal H^m\!\llcorner P$ with locally integrable multiplicity $\theta$. Test stationarity against vector fields tangent to $P$, compactly supported in a connected plane domain. In coordinates, \[ \int_P\theta\,\partial_i\phi=0 \quad\text{for all }\phi\in C_c^1(P) \text{ and all }i. \] Every distributional derivative of $\theta$ is zero. Mollification makes this explicit: on smaller connected domains, $\theta*\rho_\epsilon$ has zero ordinary gradient and hence is constant; passage to the limit shows that $\theta$ is almost everywhere constant. If $V$ is integral, that constant is a positive integer.

Thus choosing multiplicity two on a positive-measure Cantor subset of a plane and multiplicity one elsewhere does not yield a stationary counterexample. Rectifiability and integrality alone allow that construction, but its tangential first variation is nonzero. This is a direct test of a natural candidate that fails an actual hypothesis.

Finite unions of planes and the natural dimension barrier.

A finite sum $V=\sum_{j=1}^N Q_j|P_j|$ of affine $m$-planes with positive integer weights is stationary: the integral of the tangential divergence on each whole plane vanishes. Combine coincident planes first. Away from intersections the varifold is a fixed integer multiple of a smooth plane. The singular set is contained in the finite union of $P_i\cap P_j$, $i\ne j$. Every nonempty such intersection has dimension at most $m-1$, so its $\mathcal H^m$ measure is zero.

This is a complete family with genuine crossing singularities. Two transverse hyperplanes in $\mathbb R^{m+1}$ meet in an $(m-1)$-plane, showing why stationarity alone permits codimension-one singular sets. It would be incorrect to import the much stronger singular-set bounds of minimizing currents into this problem.

Nor can one immediately obtain a positive-measure singular set by taking infinitely many whole planes with integer weights through a compact region. If each such plane meets a fixed ball $B_r$, its intersection with $B_{2r}$ contains an $m$-ball of radius at least $r$. Each contributes at least $\omega_mr^m$ mass. Infinitely many distinct summands therefore violate local finite mass. A more complicated construction would need to evade this simple mass obstruction.

Dropping integrality produces a sharp failure test.

Enumerate distinct rational heights $a_j$ dense in $(-1,1)$, and let \[ V=\sum_{j\geq1}2^{-j} \left|\mathbb R^m\times\{a_j\}\right| \quad\text{in }\mathbb R^{m+1}. \] This is a rectifiable varifold with locally finite mass: on a compact ball, each plane contributes at most the volume of an $m$-ball of the same radius, and the weights sum to one. Each summand is stationary, and the absolute summability permits interchange with the first-variation integral, so $V$ is stationary.

Its support is the entire closed slab $\mathbb R^m\times[-1,1]$. No point in the open slab has a neighborhood supported on one smooth embedded $m$-submanifold. Thus the singular set contains an open $(m+1)$-dimensional region, with nonzero indeed locally infinite $\mathcal H^m$ measure. The example fails the problem's integrality hypothesis because the multiplicities are $2^{-j}$. It is not a counterexample to the catalogue assertion. It explains exactly why the preceding integer mass obstruction matters.

The same construction in higher codimension is obtained by adjoining zero coordinates. No conclusion about the integral case follows from this extension.

Why an open dense regular set is insufficient.

The regular set is relatively open by definition. Its density in the support is a known regularity consequence discussed in [S1]. Neither topological fact gives the required measure statement. A closed nowhere-dense fat Cantor subset of an interval has positive Lebesgue measure, while its complement is open and dense. Products give the same distinction in dimension $m$. These are measure-theoretic tests, not examples of stationary integral varifolds with such singular sets.

A successful argument must exploit more than the ability to find one smooth neighborhood in every ball. It would need a quantitative statement showing that smooth neighborhoods exhaust almost all of the mass, or a regularity mechanism at almost every higher-multiplicity planar tangent. The local constant-multiplicity proposition above is one sufficient mechanism, but stationarity has not been shown here to imply its hypothesis.

Verification, first gap, and outcome.

The examples were checked directly in the first-variation formula; none relies on an orientation or current boundary. The infinite planar sum has finite mass but deliberately noninteger weights. The integer planar calculation accounts for coincident planes and for their intersections. The distinction between mass measure, support and Hausdorff measure remains explicit throughout.

The first missing step is higher-multiplicity regularity on a set of full $m$-dimensional measure under stationarity alone. Tangent-plane existence almost everywhere and integrality do not by themselves supply open smooth neighborhoods. UNRESOLVED. The density-one and locally constant-multiplicity cases follow from the stated regularity input, and the elementary countertests identify failures of weaker hypotheses. No general null-singular-set theorem or integral counterexample is claimed.

Sources

• S1 Brena, Decio and De Lellis, Remarks and Conjectures on Stationary Varifolds $2025$. https://arxiv.org/html/2503.00651. Formulation source.