1 · The essentials
When must a curved surface have the shape of a simple bowl? These manuscripts study smooth surfaces that bend strictly upward locally and satisfy a particular equation governing their shape. In dimensions three through nine, a graph meeting these conditions must be an elliptic paraboloid—a bowl described by a positive quadratic expression—provided its ordinary distance measurement is complete. Completeness is a condition that rules out missing endpoints at finite distance. In dimension ten, the main manuscript constructs a smooth graph defined everywhere that satisfies the same equation and bends strictly upward, but is not quadratic. This makes dimension nine the last dimension in the stated classification range. A related classification also covers surfaces not initially assumed to be graphs, under a different completeness condition. The main result has a Lean formalization, although the supplied material does not specify which individual claims it covers.
A picture to help (an imperfect image, not to be taken literally)
As an imperfect image, imagine a rule for shaping a bowl that, under suitable conditions, permits only stretched versions of a standard quadratic bowl. The manuscripts say that this restriction holds in dimensions three through nine, while dimension ten permits a smooth alternative; the image cannot capture the equation or the completeness conditions.
Why it matters
The results identify a sharp limit for the stated Euclidean-complete graph classification: dimensions three through nine admit only elliptic paraboloids, but dimension ten admits a smooth nonquadratic entire example. The companion result also shows that an initial graph assumption can be replaced by affine completeness for a specified class of immersed surfaces. Keeping the two completeness conditions distinct is essential to understanding what the counterexample does and does not establish.
2 · Going further
The affine Bernstein question here asks whether an affine-maximal graph must be an elliptic paraboloid. For a graph represented by a smooth function , the local convexity requirement is expressed by a positive-definite Hessian . Manuscript 2 shows in the manuscript that, for graph dimensions , smoothness, local uniform convexity, classical affine maximality, and completeness of the induced Euclidean metric force the paraboloid conclusion.
Manuscript 1 supplies the contrasting construction at : a smooth nonquadratic function defined on the entire domain whose graph solves the classical affine maximal equation and has positive definite everywhere. The repository summary presents this example as making the dimension range sharp. Crucially, positivity is pointwise: the manuscript does not assert a single positive lower bound for the Hessian valid everywhere, nor does it assert completeness of the Berwald–Blaschke metric.
Manuscript 2 also treats connected, smooth, open, locally uniformly convex immersed hypersurfaces, where “open” means noncompact and without boundary. In dimensions three through nine, classical affine maximality and completeness for the affine Berwald–Blaschke metric give the same paraboloid classification, without an initial graph assumption. The two manuscripts thus pair a classification with a construction outside its dimension range. Their abstracts do not describe the argument or construction in enough detail to explain its mechanism; the supplied verification information states that the main result has a Lean formalization but does not delineate its coverage.
Useful notions
- Graphs of functions of several variables
- Second derivatives and the Hessian
- Positive-definite matrices
- Distance metrics and completeness
- Immersed hypersurfaces
What was known before / what is new
Before
The source identifies the classical affine maximal equation and calls the Euclidean-complete classification question the affine Bernstein conjecture. It also refers to an entire-graph affine Bernstein assertion. It supplies no earlier dimension ranges, historical chronology, or details of prior partial results.
New
Manuscript 2 shows in the manuscript the elliptic-paraboloid classification for Euclidean-complete graphs in dimensions three through nine. It also gives the corresponding classification for affine-complete immersed hypersurfaces under its stated hypotheses. Manuscript 1 constructs a smooth, entire, nonquadratic affine-maximal graph in dimension ten with positive-definite Hessian everywhere.
Limits and assumptions
The graph classification requires smoothness, local uniform convexity, classical affine maximality, induced Euclidean completeness, and dimension three through nine. The non-graph version requires a connected smooth open immersed hypersurface, local uniform convexity, classical affine maximality, and completeness for the affine Berwald–Blaschke metric in the same dimensions. The dimension-ten construction has pointwise positive-definite Hessian, but no global uniform lower bound or Berwald–Blaschke completeness is asserted, so it should not be presented as an affine-complete counterexample. The main result has a Lean formalization; the supplied material does not identify its precise scope or establish formal verification of every companion claim. The source does not specify what remains open outside these statements.
Mini glossary
- Elliptic paraboloid —
- A bowl-shaped surface described by a positive-definite quadratic expression.
- Entire graph —
- A graph of a function defined on the whole Euclidean domain rather than just a smaller region.
- Positive-definite Hessian —
- A matrix of second derivatives whose quadratic expression is positive in every nonzero direction at the point in question.
- Affine maximal graph —
- A graph satisfying the classical affine maximal equation studied in these manuscripts.
- Metric completeness —
- The property that points becoming arbitrarily close to one another in the chosen distance measurement converge to a point within the space.
- Berwald–Blaschke metric —
- The affine metric used for the completeness hypothesis in the companion hypersurface classification.
Keywords : Affine Bernstein rigidity · Affine maximal graphs · Metric completeness
AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.
3 · Technical details and sources
Claim made in the manuscript (source text)
Proves that every smooth locally uniformly convex affine-maximal graph of dimension three through nine, complete for its induced Euclidean metric, is an elliptic paraboloid. A smooth entire nonquadratic example in dimension ten makes this range sharp. In dimensions three through nine, the paraboloid classification also holds for connected open locally uniformly convex affine-maximal hypersurfaces complete for the affine Berwald–Blaschke metric.
Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.
Manuscripts in this family
A Smooth Nonquadratic Entire Affine Maximal Graph in Dimension Ten
05/10/2026We construct a smooth nonquadratic entire graph in dimension ten that solves the classical affine maximal equation and has positive-definite Hessian everywhere. This gives a smooth counterexample to the entire-graph affine Bernstein assertion in dimension ten. Hessian positivity is pointwise; no global uniform lower bound or completeness of the Berwald–Blaschke metric is asserted.
The affine Bernstein theorem in dimensions three through nine
24/09/2026We prove the Euclidean-complete affine Bernstein conjecture in dimensions three through nine: a smooth locally uniformly convex affine maximal graph is an elliptic paraboloid whenever its induced Euclidean metric is complete. The same conclusion holds, without an initial graph assumption, for connected smooth open (noncompact and without boundary) affine-complete locally uniformly convex immersed hypersurfaces that are classically affine maximal in these dimensions.