Singular sets of stationary integral varifolds

Source and verification

This article explains family 346 of the October 6, 2026 OpenAI collection. Research conclusions are attributed to its manuscripts. The review covers statements, introductions, and selected proof mechanisms; it is not an independent verification of the complete proofs.

Overview

A stationary integral varifold is a weak minimal surface that permits integer multiplicity and singular intersections. This family contains an almost-everywhere regularity manuscript and a later sharp bound of $m-1$ for the Hausdorff dimension of its singular set in Euclidean space. Both must address higher-multiplicity planar tangents, where a flat first approximation does not guarantee a smooth neighborhood. The shared ingredient is a signed comparison of projected mass and tilt; the later paper adds a rescaling argument that retains many singular centers. (OpenAI, 2026) (OpenAI, 2026b)

🏷️ Varifolds, stationarity, and regular support

Let $U\subset {\mathbb{R}}^{m+n}$ be open. A countably $m$-rectifiable set $E$ can be covered, apart from an $m$-dimensional Hausdorff-null set, by countably many Lipschitz images of subsets of ${\mathbb{R}}^m$. It has an approximate tangent $m$-plane at almost every point.

An integral varifold assigns a positive integer multiplicity $\theta (x)$ to this geometry. Its weight measure is

$$\mu =\theta {\mathcal{H}}^m\upharpoonright E.$$

The varifold also records the tangent planes, allowing first variation to remain meaningful under weak limits. Write $P_{T_{x}V}$ for the orthogonal projection onto the approximate tangent plane.

Stationarity means that the first derivative of area vanishes under every compactly supported ambient deformation. Equivalently,

$$\int \mathrm{tr}(P_{T_{x}V}DX)d\mu =0$$

for every compactly supported $C^1$ vector field $X$ in $U$. For a smooth submanifold, this is the minimal-surface equation. Stationarity alone does not require local area minimization or nonnegative second variation.

A support point is regular when, throughout some neighborhood, the varifold equals $q|\Sigma |$ for one positive integer $q$ and one smooth embedded minimal submanifold $\Sigma$ without boundary in that neighborhood. All remaining support points form $\mathrm{Sing}V$.

The neighborhood requirement matters. A countable smooth covering of almost all the mass does not exclude additional sheets or topology accumulating nearby. Regularity concerns the entire local support.

🏷️ The two manuscripts and the sharp bound

Results reported by family 346

For arbitrary $m,n\ge 1$, the September manuscript proves

$${\mathcal{H}}^m(\mathrm{Sing}V)=0$$

for stationary integral varifolds in Euclidean open sets. The October manuscript strengthens the conclusion to

$${\dim }_{\mathcal{H}}\mathrm{Sing}V\le m-1.$$

Neither assertion assumes stability, minimizing behavior, orientability, a fixed density upper bound, or codimension one. (OpenAI, 2026a) (OpenAI, 2026b)

The Hausdorff dimension of a set is the threshold exponent above which its Hausdorff measure vanishes. An ${\mathcal{H}}^m$-null set can still have dimension $m$, so the second assertion does not follow formally from the first.

The dimension bound is sharp. Take two distinct $m$-planes meeting along an $(m-1)$-plane, each with multiplicity one. Their sum is stationary by linearity of first variation. Away from the intersection the support is locally one plane; on the intersection it is not one embedded submanifold. Its singular set therefore has dimension exactly $m-1$.

For $m=1$, the theorem bounds singular dimension by zero; it does not state that the singular set is discrete. The Euclidean ambient hypothesis is retained throughout the main theorems. The earlier paper separately proves an almost-everywhere conclusion for full round spheres by a cone argument.

Manuscript	Main conclusion	Role in the family
September: almost-everywhere regularity	Singular set is ${\mathcal{H}}^m$-null	Signed excess, minimal fitting, inward height comparison
October: codimension-one bound	Singular dimension is at most $m-1$	Fits at prescribed scales, retained density centers, a second dimension reduction

🏷️ Tangent cones and the higher-multiplicity difficulty

Stationary monotonicity says that the mass ratio

$$\frac{\mu (B_r(p))}{{\omega }_m{r}^m}$$

is nondecreasing in $r$ while the ball remains inside $U$. Its limit as $r\downarrow 0$ is the density ${\Theta }_V(p)$. Here ${\omega }_m$ is the volume of the unit ball in ${\mathbb{R}}^m$.

Translate $p$ to the origin and dilate space by $1/r$. Varifold mass is simultaneously divided by $r^m$. Subsequential limits are tangent cones. At almost every rectifiable point the limit is $Q|\Pi |$, an $m$-plane with integer multiplicity $Q$.

Allard’s multiplicity-one regularity is a classical input. It does not automatically resolve a planar tangent with $Q>1$: dividing the whole varifold by $Q$ may destroy the needed lower density bound at neighboring points. Nor does flatness to first order exclude several sheets that approach one another more rapidly than linearly.

Classical dimension reduction already controls points with no planar tangent by dimension $m-1$. The new work must retain information beyond the leading planar limit at the remaining points.

🏷️ Projected mass, tilt, and their signed cancellation

First use a flat reference plane ${\mathbb{R}}^m\times \{0\}$ and let $\pi$ project onto it. The vertical block $P_{vv}$ of the tangent projection measures departure from horizontal tangency. Define base measures

$$\begin{array}{rl}M& ={\pi }_{\#}\mu -Qdy,\\ D& ={\pi }_{\#}(\mathrm{tr}{P}_{vv}\mu ).\end{array}$$

$M$ is a signed projected mass defect; $D$ is nonnegative tilt. The reference integer $Q$ is selected through projected counts and smooth mass averages. It is not assumed to bound the pointwise multiplicity everywhere.

The leading comparison is for $2M-D$, with its sign retained. For a small graph, area increases quadratically in the gradient, while tilt has twice that leading coefficient. An absolute-value estimate would discard the cancellation between these terms.

In a normalized flat cylinder, put $\tau =|M|+D$ and suppose $\tau (B_4)\le d=e^{-T}$. The September paper proves, for nonnegative supported Lipschitz tests with fixed bounds,

$$\int wd(2M-D)\le d\exp [-F_Q(T)].$$

The explicit rate satisfies $F_Q(T)=T^{o(1)}$ and grows faster than every fixed power of $\log T$. Its particular iterated-exponential expression is less important here than the uniformly small one-sided error. (OpenAI, 2026b)

For a curved reference graph the theorem includes a geometric error measure $\Lambda$ and compares $2M-D-C_Q\Lambda$. It also requires the physical chart radius and physical support height to be small relative to a power of the excess budget. These errors cannot be improved merely by renaming the coordinates of a smaller subchart.

The estimate is proved by induction on $Q$. A positive violation is converted by a heat-potential argument into mass assigned to selected centers and scales. If representative normal heights separate into groups, the proof obtains actual stationary restrictions with smaller reference counts and applies induction to them. At most $Q-1$ mergers of height groups occur, leaving an interval of scales without mergers.

On such an interval, comparison with a squared-distance moment forces tilt at many scales. Normal stationarity converts tilt into a height-flux estimate. A higher height moment ensures that the flux cannot all hide in tiny regions of high excess density. A dyadic packing lemma then bounds the mass of the union of full innermost witness cubes. Finally a charging estimate places the original contact mass under that small bound, contradicting the assumed violation. Keeping full cubes and cumulative scale inequalities is important; the argument does not assume a favorable pointwise sign at every larger scale.

🏷️ Almost-everywhere regularity from a fixed center

At a typical point, tangent-plane convergence and differentiation of multiplicity sets give an integer $Q$ with

$$r^{-m}\mu (B_r\cap \{\theta \ne Q\})⟶0.$$

The September manuscript fits smooth minimal graphs at successive scales, stopping when a prescribed support-height bound fails. Centered first-moment estimates and elliptic correction make nearby fits agree to higher accuracy than their original height errors.

A failed fit forces separated projection entries of lower multiplicity on a definite portion of the base. The displayed typical-point condition therefore makes the regions influenced by failures occupy a vanishing fraction of small base balls. Compatible fits are glued to one fixed center $S$ of sufficiently high finite regularity. The support height is zero outside the influence region, and the mean curvature of $S$ and its first derivative are controlled by height errors.

The remaining problem is to rule out nonzero height concentrated inside that small region. Signed excess enters again at finite radii relative to the fixed center. A separate radial estimate treats the outer collar so that the leading tilt coefficient is not lost. The resulting inward frequency comparison gives height doubling without assuming it in advance.

Along selected radii, let $H(r)>0$ denote the weighted squared-height quantity. The proof has a lower bound for the inner normalized squared-height integral by $cH(r)$ and, for some fixed $p>2$, the higher-moment bound

$$r^{-m}{\int }_{B_{r/4}}|z{|}^{p}d\mu \le CH(r{)}^{p/2}.$$

Here the ball is in center coordinates and the integration is over the varifold above it. Scalar mass control turns the vanishing base-volume fraction of the influence region into a vanishing projected-mass fraction, denoted ${\epsilon }_r$. Since height vanishes elsewhere, Hölder’s inequality gives

$$\begin{array}{c}{r}^{-m}{\int }_{B_{r/4}}|z{|}^{2}d\mu \\ \le CH(r){\epsilon }_r^{1-2/p}=o(H(r)).\end{array}$$

This contradicts the lower bound. The support must lie on the center in a neighborhood. Stationarity then gives constant integer multiplicity and makes the center smoothly minimal.

This proves a $\mu$-almost-everywhere conclusion first. To include all support points in the Hausdorff-null assertion, monotonicity gives a lower mass bound $\mu (B_r(p))\ge {\omega }_m{r}^m$ at every support point. A disjoint-ball covering argument shows that a $\mu$-null subset of the support is also ${\mathcal{H}}^m$-null. (OpenAI, 2026b)

🏷️ Flat scales carrying many singular centers

For the stronger result, suppose instead that

$$m-1<s<{\dim }_{\mathcal{H}}\mathrm{Sing}V.$$

The October paper first chooses one singular origin, an integer $Q$, a plane $\Pi$, and scales $L_i\downarrow 0$ at which the rescaled varifolds tend to $Q|\Pi |$. At these same scales, it retains nonzero measures ${\sigma }_i$ on singular points of density exactly $Q$, with

$$\begin{array}{c}{\sigma }_i(B_r(x))\le C{r}^s,\\ c\le {\sigma }_i({\overline{B}}_1)\le C.\end{array}$$

Such a ball-growth estimate is often called a Frostman bound. A nonzero limiting measure with exponent $s>m-1$ cannot be supported on an $(m-1)$-plane: covering a bounded part of that plane by $O({\epsilon }^{-(m-1)})$ balls would make its total mass at most $C{\epsilon }^{s-m+1}\to 0$.

The first dimension reduction uses monotonicity equality at retained centers to produce translation invariance of a tangent cone along at least $m$ directions. Local finiteness of its $m$-dimensional support forces it to be a plane, and integrality then identifies $Q$. Density is not assumed to be integer at every original singular point.

The subsequent fitting construction must work at the prescribed scales $L_i$; replacing them by unrelated favorable scales could lose the center measures. The paper proves that its controlled fitting intervals cover every sufficiently small target scale, including transitions between intervals.

🏷️ A nonzero centered height limit

After subtracting fitted reference graphs $S_i$ and rescaling height by $a_i\downarrow 0$, let $z_i$ be normal displacement. The construction retains three properties:

• The normalized second-height moment stays bounded and bounded away from zero on fixed interior regions.
• The normalized first-height moment tends weakly to zero.
• Mean curvature of $S_i$ and its first normal derivative are $o(a_i)$.

The first prevents the entire limit from being subtracted away. The second removes an arbitrary translation component. The third makes geometric errors disappear at the height scale. Also $|z_i|\le C{a}_i$ on the support and the mass-defect and tilt measures have size $O(a_i^2)$.

Divide the second-order measures by $a_i^2$ and pass to weak limits on $\Pi$. Denote squared height by $\nu$, mixed height-tilt flux by $f$, tilt stress by a symmetric matrix measure $\beta$, and signed mass defect by $\mathfrak{m}$. Set $d_0=\mathrm{tr}\beta$.

The limit satisfies

$$\left(\begin{array}{cc}\nu & f^{\mathsf{T}}\\ f& \beta \end{array}\right)\ge 0,\nu \ne 0,$$

and, in distributions,

$$\begin{array}{c}\nabla \nu =2f,\mathrm{div}f=d_0,\\ \mathrm{div}\beta =\nabla \mathfrak{m},\\ 2\mathfrak{m}\le d_0,|\mathfrak{m}|\le C{d}_0.\end{array}$$

Matrix positivity means that contraction with every vector gives a nonnegative measure. It is a measure-valued Cauchy–Schwarz inequality.

For intuition, a smooth harmonic map $u$ gives the same identities with $\nu =|u{|}^{2}dy$, $f_a=u\cdot {\partial }_{a}udy$, ${\beta }_{ab}={\partial }_{a}u\cdot {\partial }_{b}udy$, and $\mathfrak{m}=|Du{|}^{2}dy/2$. The actual limit is not assumed to arise from a single-valued map or a globally labeled collection of sheets.

Passing signed excess to this limit requires care. The paper first chooses new physical units sufficiently slowly that the curved error becomes $o(a_i^2)$, then covers the base by cells with radii equal to a small power of $a_i$. This verifies the local geometric and excess hypotheses without assuming a rate relating $a_i$ to the original $L_i$.

🏷️ Retaining density through the second-order normalization

Ordinary radial monotonicity on a curved reference graph has an error that can dominate $a_i^2$. The manuscript instead measures radius along the reference graph and weights its radial vector field by the inverse reference-volume density. This cancels the error of the reference geometry exactly at leading order.

Averaging the remaining error against the retained center measures introduces

$${\int }_0^l{t}^{s-m}dt<\infty .$$

Its convergence is precisely the condition $s>m-1$. This is the point where the claimed dimensional threshold enters the analysis. The estimate holds down to zero radius, without assuming that radius is larger than $a_i$.

The vanishing first-height moment removes the translation cross term. The resulting limit inequality holds at every retained center $x$:

$$2l^2\int {\varphi }_{x,l}d\mathfrak{m}\ge \int {\psi }_{x,l}d\nu .$$

Here, for one fixed integer $p\ge 4$, the compactly supported radial kernels are

$$\begin{array}{c}\chi =(1-|y-x{|}^2/l^2{)}_{+},\\ {\varphi }_{x,l}(y)=l^{-m}{\chi }^p,\\ {\psi }_{x,l}(y)=2p{l}^{-m}{\chi }^{p-1}.\end{array}$$

This density inequality is additional information beyond the linearized first-variation equations. Without it, a nonzero harmonic-map model would give no contradiction.

🏷️ Frequency and the final dimension reduction

Fix a center, write $v=y-x$ inside the integrals, and define

$$\begin{array}{rl}H(l)& =\int \psi d\nu ,\\ A(l)& =l^2\int \varphi d\mathfrak{m},\\ D(l)& =l^2\int \varphi d{d}_0,\\ R(l)& =\int \psi (y)d\beta (y)[v,v].\end{array}$$

Dots below denote derivatives in $\log l$. Testing the distributional equations against the radial kernels gives

$$\begin{array}{c}\dot{A}=2A-D+R,\dot{H}=2D,\\ RH\ge D^2,2A\le D.\end{array}$$

The quadratic inequality is Cauchy–Schwarz for the positive matrix measure. A finite-radius continuation argument proves $H(l)>0$ on every interior ball: a nonzero height measure satisfying these equations cannot vanish on an open region.

Put $P=A/H$, $\eta =D/H$, and $\rho =R/H$. At retained centers, the density comparison gives $2P\ge 1$. Consequently

$$\begin{array}{c}\eta \ge 2P\ge 1,\rho \ge {\eta }^2,\\ \dot{P}=2P-\eta +\rho -2P\eta \\ \ge (\eta -1)(\eta -2P)\ge 0.\end{array}$$

The frequency is therefore monotone at all those centers and has a finite limit inward.

A second rescaling retains a nonzero measure of centers spanning the whole base plane and makes the limiting frequency the same constant $P_0\ge 1/2$ at those centers and at the origin. This uses uniform convergence of the frequency on a suitable compact subset of centers, together with the Frostman bound and weak compactness.

Equality in the differential inequality forces $\eta =2P_0=:{\eta }_0\ge 1$ and $\rho ={\eta }_0^2$. Contract the positive matrix measure with the vector $(-{\eta }_0,y-x)$. Its weighted integral is

$${\eta }_0^{2}H-2{\eta }_{0}D+R=0.$$

A positive semidefinite matrix whose quadratic form vanishes on a vector annihilates that vector. The first row therefore gives, as a measure identity,

$$(y-x)\cdot d{f}_{\infty }(y)={\eta }_{0}d{\nu }_{\infty }(y).$$

Subtract the same identity at the origin. For every retained center, $x\cdot f_{\infty }=0$. Since those centers span the base plane, $f_{\infty }=0$, and the identity at zero forces ${\nu }_{\infty }=0$. This contradicts its normalized positive height. Hence a set of retained centers cannot have dimension above $m-1$, completing the sharper theorem. (OpenAI, 2026a)

🏷️ The round-sphere consequence

The September paper treats a stationary varifold on the full round sphere by taking its Euclidean cone. The cone is an integral $(m+1)$-varifold with radial mass factor $r^{m}dr$. Radial integration by parts and spherical stationarity make it stationary off the vertex; a cutoff error tending to zero extends stationarity across the vertex for $m\ge 1$.

Away from the vertex, cone regularity is equivalent to regularity of the spherical section. The singular set is therefore the radial product of the sphere’s singular set with $(0,\infty )$. Apply the Euclidean almost-everywhere theorem to the cone and slice by radius to get the spherical ${\mathcal{H}}^m$-null conclusion. This separate geometric argument does not assert the same theorem for arbitrary curved ambient metrics. (OpenAI, 2026b)

🏷️ Key steps and proof outlines

Signed excess is the second-order information that ordinary flatness loses

For a small graph with derivative matrix $Du$, its area factor is $1+|Du{|}^2/2+O(|Du{|}^4)$, while the vertical tangent-projection trace is $|Du{|}^2+O(|Du{|}^4)$. Thus $2M-D$ cancels the quadratic graph contribution. Its one-sided sign can survive a height blow-up even when separate absolute bounds for $M$ and $D$ would leave an error of the same size as the limit.

The September manuscript proves the signed estimate by induction on the reference count, using separated height configurations to reduce that count. A scale interval without mergers then turns a hypothetical positive violation into many tilt witnesses. The height moment prevents concentration from hiding those witnesses, and packing full witness cubes controls their total contribution. The subsequent inward estimate combines that signed information with a fitted center; it does not assume a labeled multigraph representation. (OpenAI, 2026b)

The centered limit retains three different kinds of information

The October construction keeps nonzero second height, removes first height, and makes the mean curvature of the reference graph negligible at the chosen height scale. Respectively, these conditions prevent a zero limit, remove translation ambiguity, and eliminate geometric forcing from the linearized equations.

The positive matrix measure records the Cauchy–Schwarz relation between height and derivatives. The first-variation equations supply its divergence identities. The retained density-center inequality adds a separate radial lower bound. A harmonic-map-type tuple can satisfy the first two groups without contradiction, so the third cannot be omitted. Its proof averages a reference-adapted monotonicity error against a measure with ball growth $r^s$, producing the integrable factor ${\int }_0^l{t}^{s-m}dt$. The strict threshold $s>m-1$ is used here. (OpenAI, 2026a)

Frequency equality becomes a matrix annihilation identity

The radial equations give, with $P=A/H$, $\eta =D/H$, and $\rho =R/H$,

$$\dot{P}=2P-\eta +\rho -2P\eta \ge (\eta -1)(\eta -2P)\ge 0.$$

The inequalities use $\rho \ge {\eta }^2$ from positivity, $\eta \ge 2P$ from signed excess, and $2P\ge 1$ from the retained centers. After the second blow-up, $P$ is constant at those centers. Equality forces $\eta =2P_0=:a\ge 1$ and $\rho =a^2$.

Contract the limiting matrix measure with $(-a,y-x)$ and integrate against a positive radial kernel. The result is $a^{2}H-2aD+R=0$. Writing the matrix measure as a positive semidefinite matrix density relative to its trace measure, a zero quadratic form implies that the matrix annihilates this vector almost everywhere. Its first row gives

$$(y-x)\cdot d{f}_{\infty }(y)=ad{\nu }_{\infty }(y).$$

Subtract the identity at center zero: $x\cdot f_{\infty }=0$. The retained centers span the base plane, because their dimension exceeds $m-1$. Hence $f_{\infty }=0$, and the origin identity gives ${\nu }_{\infty }=0$, contradicting normalized positive height. This closes the dimension reduction using measure identities alone, without reconstructing single-valued sheets or assuming unique tangent cones at the original singular points.

🏷️ Connections

Isoperimetry and integral fillings also uses weak geometric objects and sharp dimensional statements, but filling minimality supplies structure unavailable under stationarity alone.

on min-max theory and the Willmore conjecture introduces variational production of minimal surfaces. The present family concerns local regularity of arbitrary stationary integral varifolds, without the extra stability or replacement properties arising in specific min-max constructions.

🏷️ Manuscripts and verification scope

• A Codimension-One Bound for the Singular Set of a Stationary Integral Varifold — pinned manuscript and companion source.
• Almost-everywhere regularity of stationary integral varifolds — pinned manuscript and companion source.

No family scope page was present at lean/docs/346.md in the pinned snapshot. This does not establish that all related formalization code is absent. No Lean build, axiom audit, or manuscript-to-formal-statement equivalence check was performed for these notes.

OpenAI collection provides prerequisites and the reading sequence; coverage register records completed and pending families.

🐻  OpenAI 2026a. A Codimension-One Bound for the Singular Set of a Stationary Integral Varifold.

🐻  OpenAI 2026b. Almost-everywhere regularity of stationary integral varifolds.