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Math-2651 · mathematics · Geometry

Affine Bernstein assertion in dimensions three through nine

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For $3\leq n\leq 9$, let u be smooth on a nonempty open convex $\Omega \subset R^{n}$ with $D^{2}u$ positive definite and $\Sigma _ij$ $U^ij$ $\partial _ij(det$ $D^{2}u)^(-(n+1)/(n+2))=0$, $U=det(D^{2}u)(D^{2}u)^{-1}$. If the graph metric $\delta _ij+u_i$ $u_j$ is complete, must $\Omega =R^{n}$ and $u(x)=\tfrac12 x^{T}Ax+b\cdot x+c$ with A positive definite?

Short English statement

For $3\leq n\leq 9$, let u be smooth on a nonempty open convex $\Omega \subset R^{n}$ with $D^{2}u$ positive definite and $\Sigma _ij$ $U^ij$ $\partial _ij(det$ $D^{2}u)^(-(n+1)/(n+2))=0$, $U=det(D^{2}u)(D^{2}u)^{-1}$. If the graph metric $\delta _ij+u_i$ $u_j$ is complete, must $\Omega =R^{n}$ and $u(x)=\tfrac12 x^{T}Ax+b\cdot x+c$ with A positive definite?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2651-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2651

Affine Bernstein rigidity through dimension nine: The paper claims that Euclidean-complete smooth strictly convex affine-maximal graphs in dimensions three through nine are elliptic paraboloids. It also treats affine-complete immersed hypersurfaces without an initial global graph hypothesis. The parameters, regularity assumptions and normalization in this question are retained in applying that assertion.

With the hypotheses and conventions above, the claimed result addresses the whole mathematical assertion on this page.

Affine Bernstein assertion in dimensions three through nine

OpenAI's claim

Affine Bernstein rigidity through dimension nine: The paper claims that Euclidean-complete smooth strictly convex affine-maximal graphs in dimensions three through nine are elliptic paraboloids. It also treats affine-complete immersed hypersurfaces without an initial global graph hypothesis.

Mathematical statement of the claimed result

Mathematical claim 1.
For $3\le n\le9$, take a convex open $\Omega\subset\mathbb R^n$ with $\Omega\ne\varnothing$ and $u\in C^\infty(\Omega)$ have positive-definite Hessian and solve \[\sum_{i,j}U^{ij}\partial_{ij}(\det D^2u)^{-(n+1)/(n+2)}=0,\quad U=\det(D^2u)(D^2u)^{-1}.\] If $g_{ij}=\delta_{ij}+u_iu_j$, the Euclidean metric induced on that graph, is complete, the claimed conclusion is $\Omega=\mathbb R^n$ and $u(x)=\tfrac12x^{\mathsf T}Ax+b\cdot x+c$ with positive-definite symmetric $A$. The completeness hypothesis concerns the induced Euclidean graph metric.

Outline of the argument

Affine Bernstein rigidity through dimension nine: Tangent sections are analyzed through affine limits of the epigraph. Maximizing the number of nonnegative model directions yields uniform centering of transverse fibers, since a centering failure would create a further direction after rescaling. This section balance supplies the global control needed for the affine equation’s rigidity conclusion, with a separate immersion reduction handling the nongraph setting.

Formal verification

Family 353 describes Lean coverage for the Euclidean-complete graph rigidity in dimensions three through nine; the nongraph and dimension-ten assertions are separate.

Affine Bernstein rigidity through dimension nine: The manuscript links AffineBernstein.lean.

The documented selection includes this manuscript’s main assertion; that assertion retains the restrictions stated above.

This site has not run the Lean development. These links describe OpenAI’s selected formal statements and should be read with their stated scope.

Status caveat

OpenAI reports that an internal model produced the manuscripts. Their arguments have not been independently verified as proofs by this site. Any portion outside the documented formal scope may contain mistakes, and the catalogue label reports the announced mathematical claim.

Sources

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