How large can the singular set of a stationary surface be?

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Family 346 · Differential geometry
Importance 5/5 · Provisional editorial rating
Sources consulted 2026-10-26

1. The question

A smooth minimal surface has zero first-order change of area under every small deformation supported away from its boundary. But limits of minimal surfaces need not stay smooth: sheets can merge, cross, or accumulate. Geometric measure theory uses varifolds to retain area and tangent-plane information through such limits.

The basic question is how much of a stationary varifold must still have a genuinely smooth neighborhood. A tangent plane almost everywhere is not enough. Several sheets could be nearly coincident, and singular points could accumulate near a point that looks planar under magnification.

Family 346 contains two related claims. The September manuscript says that the singular set has zero measure in the dimension of the varifold. The October manuscript strengthens this in Euclidean space to a sharp Hausdorff-dimension bound: an $m$-dimensional stationary integral varifold has singular set of dimension at most $m-1$.

No area-minimizing or stability assumption is imposed. The family therefore concerns the weakest usual variational notion of a minimal surface, in every positive dimension and codimension.

2. Background and essential definitions

A surface with multiplicity and tangent information

An integral $m$-varifold in an open set $U\subset\mathbb R^{m+n}$ is built from a countably rectifiable set $E$, with a positive integer multiplicity $\theta$. Rectifiability means that, outside a set of zero $m$-dimensional measure, the set can be covered by countably many Lipschitz images of subsets of $\mathbb R^m$.

Its weight measure is

$$\mu(A)=\int_{E\cap A}\theta\,d\mathcal H^m,$$

where $\mathcal H^m$ is $m$-dimensional Hausdorff measure, the extension of length, area, and volume to irregular sets. The varifold also records the approximate tangent $m$-plane at almost every point. It does not require a choice of orientation.

Multiplicity allows, for example, a plane counted twice. Local finiteness of the measure is required, but the theorems impose no fixed upper bound on multiplicity.

Stationary is weaker than minimizing

Stationarity means that the first derivative of area vanishes under every compactly supported ambient deformation. If $P_x$ is orthogonal projection onto the approximate tangent plane and $X$ is a compactly supported vector field, this is

$$\int\operatorname{tr}(P_xDX(x))\,d\mu(x)=0.$$

The trace is tangential divergence: the infinitesimal rate of area change along the surface. On a smooth submanifold, stationarity is equivalent to zero mean curvature.

A stationary surface may still have deformations that decrease area to second order. Stability excludes those negative second variations, and area minimization is stronger still. Neither stronger condition is assumed here.

What counts as a singular point?

A support point is regular if an entire neighborhood contains exactly one smooth embedded minimal submanifold, counted with a constant positive integer multiplicity. Every other support point is singular.

A double plane is regular under this definition. Two distinct crossing planes are singular along their intersection, because their union is not one embedded submanifold there. Smooth parametrization of most of the mass is insufficient if extra pieces remain in every neighborhood.

Hausdorff dimension $\dim_H$ distinguishes the size of thin sets more finely than measure alone. A set can have zero $m$-dimensional measure while having dimension strictly greater than $m-1$, so the dimension theorem is stronger than almost-everywhere regularity.

3. Development history and previous partial results

Allard’s compactness and regularity theory established the foundational connection between weak convergence and smooth geometry. It gives regularity near appropriate multiplicity-one planar behavior and a relatively open dense regular set. At higher planar multiplicity, simply dividing by that multiplicity can destroy the neighborhood density hypotheses needed by the theorem. Allard, 1972.

Classical dimension reduction controls points at which no tangent cone is a plane. Naber–Valtorta strengthened understanding of these strata through rectifiability results. The remaining difficulty is singular points with higher-multiplicity planar tangents. Their blowups look flat even though their actual support may fail to be smooth nearby. Naber–Valtorta.

For area-minimizing integral currents, Almgren’s theory gives the stronger arbitrary-codimension bound $m-2$. De Lellis–Spadaro developed a new proof using multiple-valued functions, center manifolds, and frequency methods. Those results depend on minimizing hypotheses and cannot simply be applied to arbitrary stationary varifolds. Crossing planes already rule out an $m-2$ bound in the present setting. De Lellis–Spadaro, center manifolds.

Progress also came from special multiplicity and graphical classes. Hirsch–Spolaor proved the $m-1$ estimate for two-valued stationary Lipschitz graphs and almost-everywhere regularity for stationary codimension-one integral currents with density at most two. These are important predecessors but retain assumptions absent from the family’s general claims. Two-valued graphs, stationary hypercurrents.

A separate line concerns how smoothly one can cover almost all of the measure. Menne proved second-order rectifiability, and Brena–De Lellis–Franceschini proved smooth rectifiability and infinite-order approximation. Kolasiński later supplied an alternative approach. A countable smooth covering does not say that the whole support in a neighborhood equals one smooth piece; that stronger local conclusion is the issue here. Brena–De Lellis–Franceschini, Kolasiński.

Brena–Decio–De Lellis explicitly separated the measure-zero and dimension-bound conjectures. The two manuscripts in this family address those two levels. Their conjectures and background.

4. What the new manuscripts claim

Let $m,n\geq1$, let $U\subset\mathbb R^{m+n}$ be open, and let $V$ be a stationary integral $m$-varifold in $U$.

M2, September 23:

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

Thus almost every support point has a neighborhood on which the varifold is a constant positive integer multiple of a smooth embedded minimal submanifold. M2 also states this almost-everywhere conclusion for integral varifolds stationary within a full round sphere of any positive radius, in arbitrary positive dimension and codimension.

M1, October 5:

$$\dim_H\operatorname{Sing}V\leq m-1.$$

This Euclidean estimate is sharp. Take two distinct $m$-planes intersecting along an $(m-1)$-plane and add their varifolds. Each plane is stationary, so their sum is stationary. Its singular set is precisely the intersection, attaining the bound.

Both manuscripts omit stability, minimization, orientability, a prescribed density upper bound, and any codimension-one restriction. The conclusions are local inside $U$; boundary behavior outside that open set is not included.

M1 imports M2’s signed-excess theorem and local fitting and frequency estimates. It does not merely apply M2’s measure-zero conclusion, which by itself would not imply the dimension bound. These are connected stages of a program, not two independent proofs of the same theorem.

The round-sphere statement explicitly supplied by M2 is almost-everywhere regularity. The main sharp dimension theorem is stated in Euclidean open sets.

5. Why the advance matters

Importance: 5/5. If correct, the family resolves fundamental regularity questions for a very general class of stationary geometric objects. It identifies the largest possible singular-set dimension without assuming the stronger variational properties used in much of minimal-surface regularity theory.

The result would close the gap between looking smooth in a measure-theoretic covering and having an actual smooth support neighborhood. Higher-multiplicity planar tangents are precisely where that gap is hardest to bridge.

The sharpness example also clarifies what can reasonably be expected. Crossings are allowed by stationarity, so universal complete smoothness is impossible. The claimed theorem gives the optimal dimension ceiling consistent with those unavoidable examples.

The methods combine signed estimates, curved reference surfaces, and measure-valued blowups. Their significance lies in adapting tools inspired by minimizing-current theory to a setting where minimization is unavailable. This rating concerns the stated claims, not a verification of those adaptations.

6. The main idea

Preserve a cancellation between excess mass and tilt

Near a planar tangent, choose horizontal and vertical coordinates. Project the weight measure to the horizontal plane. Relative to a multiplicity-$Q$ plane, let $M$ denote the signed projected-mass discrepancy, and let $D$ measure how much the tangent planes tilt vertically.

The key quantity is

$$2M-D.$$

Both terms have leading contributions from a small graphical slope, so preserving their signs permits cancellation. Replacing everything by absolute values would discard the improvement the argument needs.

M2 claims a very small upper error for suitable weighted integrals of this signed quantity. Its proof proceeds by induction on integer multiplicity. When groups of sheets separate in height, lower-multiplicity estimates become available. Heat-potential contacts select centers and scales where a hypothetical violation can be measured, and a packing argument charges those contacts to an insufficient amount of excess mass.

Fit a curved center and force the support onto it

A flat plane is often too crude a reference. The manuscript fits smooth minimal graphs at successive scales, stopping when the support-height error is too large. Nearby successful fits agree sufficiently well to be glued into a single curved center.

A centered first-moment estimate controls average height, while second and higher moments measure spread. A frequency function compares height with derivative energy and records the order of vanishing. The signed estimate supplies an inward comparison and height doubling, rather than assuming doubling in advance.

At points where other multiplicities occupy vanishing relative mass, the failed-fit regions become too small to carry nonzero height. The support must therefore lie on the center in a neighborhood. Stationarity then makes the multiplicity constant and the center smooth and minimal. This yields M2’s almost-everywhere conclusion.

For the sphere consequence, M2 separately forms a Euclidean cone over the spherical varifold and relates stationarity and regularity away from the cone vertex. It is a geometric reduction specific to the round metric.

Retain a large set of singular centers through magnification

M1 assumes for contradiction that the singular set has dimension greater than $m-1$. Choose

$$m-1<s<\dim_H\operatorname{Sing}V.$$

Dimension reduction supplies scales converging to a multiplicity-$Q$ plane and measures $\sigma_i$ on nearby singular centers satisfying

$$\sigma_i(B_r(x))\leq Cr^s,$$

with nonvanishing total mass. These estimates express that the centers are too numerous to concentrate on an $(m-1)$-dimensional plane.

The reference graphs must be constructed at these prescribed scales, not merely along another convenient sequence. After subtracting the reference and normalizing the height, the limit must retain a nonzero second moment while its first moment tends to zero. Otherwise the normalization could erase all the geometry or leave an irrelevant translation.

The limit is described by measures of height, tilt, mixed height–tilt, and signed mass. This avoids assuming that all sheets can be globally labeled as ordinary functions.

The exponent appears in a radial error integral

Curved reference graphs introduce errors into the comparison at each singular center. The paper corrects radial test fields by the inverse reference-volume density and averages their errors against the center measures. The decisive integral has the form

$$\int_0^\ell t^{s-m}\,dt<\infty,$$

which holds exactly when $s>m-1$. This is where the claimed dimension threshold enters quantitatively.

A further blowup gives constant frequency at enough independent retained centers. Equality in a positive-semidefinite height–tilt inequality then forces the limiting height measure to vanish. That contradicts its nonzero normalization and yields the dimension bound.

7. Scope, limitations, and open questions

The sharp estimate uses the Euclidean ambient metric. Smooth non-Euclidean metrics admit examples with planar tangents and accumulating topology; one should not silently extend the theorem to every Riemannian ambient manifold. De Lellis–Hirsch–Lihn–Spolaor.

A Hausdorff-dimension bound does not automatically give a locally finite $(m-1)$-dimensional measure, rectifiability of the entire singular set, or uniqueness of every tangent cone. The theorem states none of these stronger conclusions. For $m=1$, a zero-dimensional bound also does not by itself mean a discrete or finite singular set.

The result permits crossing singularities and arbitrary positive integer multiplicity. It does not upgrade stationarity to minimizing behavior or improve the universal bound below the sharp crossing example.

This digest explains the claims and the dependency between the manuscripts. It does not check their signed-excess estimates, fitting constructions, or final dimension-reduction argument.

8. Sources and further reading

The primary sources linked above give the classical regularity theory, recent partial results, and the precise conjectures. Optional background appears under varifold, Hausdorff dimension, and minimal surface.

Source manuscripts

• M1. OpenAI, A Codimension-One Bound for the Singular Set of a Stationary Integral Varifold (October 5, 2026).
• M2. OpenAI, Almost-everywhere regularity of stationary integral varifolds (September 23, 2026).

Sources consulted on 26 October 2026 at repository revision 1581d4c. Download the manuscript citations.