When affine maximal graphs must be paraboloids—and when they need not be

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Family 353 · Differential geometry
Importance 5/5 · Provisional editorial rating
Sources consulted 2026-10-26

1. The question

A parabola is the simplest strictly convex curve. Its higher-dimensional relative, an elliptic paraboloid, is the graph of a positive quadratic function. Can a natural geometric equation single out these shapes among all complete convex graphs?

The affine Bernstein problem asks this for hypersurfaces that are stationary for affine area. This area measures curvature as well as ordinary size and behaves naturally under affine coordinate changes. Its stationarity equation is fourth order: it involves four derivatives of the function defining the graph.

Family 353 contains two complementary claims. M2 says that in dimensions three through nine, a smooth strictly convex graph satisfying the classical equation must be a paraboloid if it is complete in the ordinary Euclidean metric. M1 constructs a smooth, everywhere strictly convex, nonquadratic solution on all of ten-dimensional space. Taken with the classical two-dimensional result, these claims put a sharp boundary between rigidity and flexibility for smooth entire graphs at dimension ten.

The word “complete” needs care. Affine geometry supplies its own metric, different from the metric inherited from Euclidean space. The counterexample is Euclidean-complete; the paper does not assert that its affine metric is complete.

2. Background and essential definitions

Convex graphs and their equation

Let $\Omega\subset\mathbb R^n$ be an open convex set and let $u:\Omega\to\mathbb R$ be smooth. Its graph consists of points $(x,u(x))$ in $\mathbb R^{n+1}$. The graph has dimension $n$, even though the surrounding space has dimension $n+1$.

The Hessian $D^2u=(u_{ij})$ is the matrix of second derivatives. Here locally uniformly convex means that this matrix is positive definite at every point. On each compact subset its eigenvalues consequently have a positive lower bound. No single positive lower bound over the whole graph is assumed.

For a compact region of the domain, affine area is given by the integral

$$\mathcal A(u)=\int (\det D^2u)^{1/(n+2)}\,dx.$$

The determinant measures the product of the principal second-derivative contributions. A graph is classically affine maximal when the first variation of this functional vanishes under compactly supported smooth variations. Its equation is

$$\sum_{i,j=1}^n U^{ij}w_{ij}=0, \qquad w=(\det D^2u)^{-(n+1)/(n+2)},$$

where $U^{ij}$ are the entries of the cofactor matrix

$$U=(\det D^2u)(D^2u)^{-1}.$$

Although the equation looks second order in $w$, that quantity already depends on two derivatives of $u$. This is why the original equation is fourth order. “Maximal” is the established geometric terminology; the formulation used here is stationarity.

A positive quadratic polynomial has constant positive Hessian, hence constant $w$, and satisfies the equation immediately. The question is whether other complete solutions exist.

Two meanings of completeness

Distances along the graph in ordinary Euclidean space are described by

$$g_{ij}=\delta_{ij}+u_i u_j.$$

Completeness means that a path escaping every compact part of the graph cannot have finite length. An entire graph, with $\Omega=\mathbb R^n$, is complete for this metric because its lengths dominate those of its projection to $\mathbb R^n$.

The affine, or Berwald–Blaschke, metric is instead

$$g^{\mathrm{aff}}_{ij}=(\det D^2u)^{-1/(n+2)}u_{ij}.$$

Its lengths depend directly on second derivatives. Completeness for one metric cannot simply be substituted for completeness for the other.

3. Development history and previous partial results

Chern asked whether an entire locally uniformly convex affine maximal surface must be an elliptic paraboloid. Calabi developed the affine-area framework and proved a surface rigidity theorem under both Euclidean and affine completeness. These works established the geometric problem before the modern regularity machinery was available.

Trudinger and Wang’s 2000 paper proved the Euclidean-complete surface theorem without requiring affine completeness. It also developed higher-dimensional interior estimates: once suitably normalized pieces of a graph retain uniform strict convexity, derivative bounds can force global rigidity. The obstacle was obtaining that convexity control from completeness alone.

The same paper constructed the famous singular ten-dimensional example

$$u_0(x,t)=\sqrt{|x|^9+t^2},\qquad (x,t)\in\mathbb R^9\times\mathbb R.$$

This showed that unrestricted rigidity could fail for generalized convex solutions. It did not provide a smooth graph with positive-definite Hessian everywhere. Removing the singularity while preserving the classical equation was a separate problem.

Trudinger and Wang subsequently proved that an affine-complete locally uniformly convex hypersurface is Euclidean-complete and established a global graph representation. Li and Jia gave a separate proof of the affine-complete surface theorem. These results explain why graph rigidity can also yield a theorem for certain immersed hypersurfaces that are not initially presented as graphs.

Later analytic approaches included Wang and Zhou’s partial Legendre transform method, with a new surface proof and higher-dimensional results under integral assumptions. Results for modified determinant powers also appeared. Such equations resemble the classical one, but their exponent is part of the problem, not a disposable parameter.

There is an additional contemporary source: Trudinger and Wang’s The higher dimensional affine Bernstein and Plateau problems, posted on 7 October 2026, states smooth counterexamples in dimensions ten and above, interior regularity for the affine Plateau problem in all dimensions, and rigidity under a growth condition. Its arXiv record also notes a February 2026 submission. These claims deserve explicit credit; this digest does not determine priority between constructions. Family 353 is discussed in its own stated scope.

4. What the new manuscripts claim

M2: completeness forces a quadratic in dimensions three through nine

Suppose $3\le n\le9$, $\Omega$ is nonempty, open and convex, and $u\in C^\infty(\Omega)$ has positive-definite Hessian and satisfies the classical affine maximal equation. M2 claims that Euclidean completeness of its graph implies

$$\Omega=\mathbb R^n, \qquad u(x)=\tfrac12 x^{\mathsf T}Ax+b\cdot x+c,$$

where $A$ is a constant symmetric positive-definite matrix, $b$ a vector, and $c$ a constant. Thus an invertible affine transformation takes the graph to the standard elliptic paraboloid.

No growth condition at infinity, global Hessian bound, or integral estimate is assumed. The domain need not be all of space at the outset; the conclusion includes that fact.

M2 also states an affine-complete consequence. A connected, smooth, locally uniformly convex, classically affine maximal immersed hypersurface of dimension three through nine is an elliptic paraboloid if its underlying manifold is noncompact without boundary and its affine metric is complete. This uses the earlier completeness and graph theorems. It does not identify the two metrics.

M1: a smooth ten-dimensional counterexample

M1 claims a function

$$u\in C^\infty(\mathbb R^{10}),\qquad D^2u>0,$$

which solves the classical equation but is not quadratic. The construction has the form

$$u(x,t)=\sqrt{F(|x|)^2+t^2},\qquad x\in\mathbb R^9,$$

with $F$ positive, smooth as a radial function on $\mathbb R^9$, and strictly convex. Its normalization is $F(0)=1$. Along the last coordinate axis, therefore, $u(0,t)=\sqrt{1+t^2}$, which is visibly nonquadratic.

The graph is Euclidean-complete because it is entire. The theorem does not impose affine completeness or a global uniform lower bound for the Hessian. It is a smooth counterexample to the classical equation, not merely to a related determinant-power equation.

5. Why the advance matters

Importance: 5/5, conditional on the claims. The affine Bernstein problem is a central classification question linking convexity, geometric invariance and nonlinear differential equations. An unconditional rigidity theorem through dimension nine, paired with a smooth failure in dimension ten, would establish a sharp dimensional transition for entire classical solutions.

The two directions illuminate different obstacles. For rigidity, the challenge is preventing a complete graph from becoming arbitrarily poorly shaped after normalization. For flexibility, it is preserving an exact fourth-order equation while replacing the singular center of a model by a smooth cap.

The result also teaches a useful lesson about geometric PDE: a familiar-looking equation can change character when either the dimension or an exponent changes. Convexity alone does not eliminate that sensitivity. The score reflects this structural significance, not a judgment about proof validity or exclusivity relative to other 2026 work.

6. The main idea

Rigidity: control every section of the graph

At a point $a$, the tangent affine function is

$$\ell_a(x)=u(a)+Du(a)\cdot(x-a).$$

A tangent section of height $h>0$ is the region

$$S_u(a,h)=\{x:u(x)<\ell_a(x)+h\}.$$

It is the part of the domain lying beneath a plane raised a fixed amount above the tangent plane. These sections record how quickly the graph separates from its tangents.

M2 seeks a uniform balance property: reflecting a section through its base point and shrinking it by one fixed factor places it inside the original section. The same factor must work at every point and every height. This rules out increasingly one-sided sections.

The manuscript studies affine rescalings of the epigraph, the convex region above the graph. Degenerate limiting shapes are organized by the number of independent nonnegative directions they contain. Maximizing that number forces their transverse slices to stay centered: otherwise, another rescaling would create an additional direction.

The equation then excludes models with two or more such directions. In logarithmic coordinates, an integral inequality forces a relevant measure to grow exponentially, whereas geometric bounds on normalized pieces allow only polynomial growth. The crucial quadratic estimate is strictly positive in dimensions three through nine. The estimates are applied to smooth approximations on a finite large region; the limiting convex body is not assumed smooth.

With this degeneration excluded, the tangent sections remain uniformly balanced. Trudinger–Wang interior estimates apply after normalizing sections by ellipsoids. Third derivatives of the normalized functions stay bounded. Transforming back along larger and larger sections makes the original third derivatives tend to zero. Thus $D^3u=0$, exactly the condition that $u$ be quadratic.

Flexibility: replace a singular profile by a controlled radial solution

M1 starts with the square-root ansatz above, but solves for its radial factor rather than smoothing the old example arbitrarily. An auxiliary positive function $P$ converts the affine maximal equation into a radial differential system. Writing $r=|x|$ and $y=F'(r)$, the ten-dimensional system is

$$F'=y,\qquad y^8y'=r^8F P^{-12/11},$$

$$(y^8P')'=-11r^8P^{-1/11}.$$

The powers reflect the nine radial coordinates and the classical ten-dimensional exponent. Positivity of $F,P,y,y'$ is essential: it keeps the equation meaningful and the constructed graph strictly convex.

The origin looks singular in radial coordinates. The manuscript first uses the variable $r^2/2$ and an integral formulation to build a smooth local solution with $F(0)=P(0)=1$. The remaining danger is that this solution breaks down at a finite radius, for example because $P$ reaches zero.

Logarithmic derivative variables turn the continuation problem into a three-dimensional dynamical system. The paper constructs a bounded box whose faces cannot be crossed outward. Its upper corner is suggested by the old singular profile. Staying inside the box controls the growth and decay of the original functions on every finite radial interval, allowing continuation to all radii.

Thus the singular example guides the large-scale bounds without being used as the final graph. Smoothness at the center and control at infinity are separate parts of the construction.

7. Scope, limitations, and open questions

These are claims about smooth, locally uniformly convex solutions of the classical affine maximal equation. Weak solutions, graphs with degenerate Hessian, and equations with another determinant power require separate statements.

The dimensional threshold refers to the dimension of the graph. M1’s example lies in $\mathbb R^{11}$ and is a graph over $\mathbb R^{10}$. Its theorem is a dimension-ten construction. Extending it by simply adding quadratic variables is not automatic, because changing dimension changes the classical exponent.

The Euclidean-complete counterexample does not resolve the corresponding question under affine completeness. Conversely, M2’s affine-complete corollary has explicit noncompactness, connectedness and convexity assumptions. Neither paper supplies a classification of all high-dimensional nonquadratic solutions.

Natural further questions concern their growth, asymptotic geometry, affine-metric completeness and the extent to which other symmetry classes allow similar constructions. The contemporary Trudinger–Wang paper provides additional results in this direction; comparison of the proofs or their correctness is outside this exposition. The mathematical claims here are reported as stated.

8. Sources and further reading

M1 gives the radial smooth counterexample; M2 gives the low-dimensional rigidity theorem and its affine-complete consequence. Their introductions distinguish the completeness assumptions and related determinant-power equations.

Historical sources include Trudinger–Wang, The Bernstein problem for affine maximal hypersurfaces, and their Affine complete locally convex hypersurfaces. Their 2026 paper is relevant contemporary work. For the separate range of generalized exponents, see Sun, Xing and Xu.

Source manuscripts

• M1. OpenAI, A Smooth Nonquadratic Entire Affine Maximal Graph in Dimension Ten (October 5, 2026).
• M2. OpenAI, The affine Bernstein theorem in dimensions three through nine (September 24, 2026).

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