A mathematical preprint reports that Euclidean-complete affine maximal type hypersurfaces that are not elliptic paraboloids exist for n ≥ 2 across the parameter interval a ∈ [−n/(n+1), 0). In the notation of the result, n sets the dimension threshold and a is the parameter whose range is being extended. That is the central question of the work.
The objects are graph hypersurfaces, or geometric graphs, associated with locally uniformly convex solutions on domains in Euclidean space. The paper examines whether these graphs have the required affine maximal type property and whether they are Euclidean complete.
A wider interval, with a narrower explicit construction
For the explicit family in Theorem 1.2, the authors report Euclidean-complete affine maximal type hypersurfaces when n ≥ 2 and a lies in the open interval (−n/(n+1), −(n−1)/n). The broader existence statement in Theorem 1.3 covers the half-open interval [−n/(n+1), 0), including examples that are not elliptic paraboloids.
That distinction matters because the constructed family and the full-range conclusion do not have identical parameter bounds. The paper presents a direct family on the open interval, while its main existence statement covers the wider interval and answers the range-extension problem.
To establish the constructed family, the proof computes the Hessian, the matrix of second derivatives, along with its determinant and inverse. The resulting expressions are substituted into the affine maximal type equation, followed by a verification of Euclidean completeness. The reported conclusion comes from these direct calculations and checks.
A conditional map of two-dimensional surfaces
The paper also turns to a classification problem in two-dimensional Calabi affine geometry. Its method uses local frame fields and an analysis of the Codazzi equations, which impose consistency conditions on the local geometric data.
A central condition is the identity in equation (4.1), with c ≠ 0 and nonconstant |T|. Under the stated two-dimensional assumptions, including A_TT ∥ T, Theorem 4.2 gives three Calabi-affine equivalence types.
In a separate flat-surface result, Theorem 4.1 classifies the specified flat Calabi affine maximal surfaces into eight Calabi-affine equivalence types, with the types stated as open parts.
The paper highlights a more specific extension within type (iii) in the two-dimensional case. It says that this type is Euclidean complete for a ∈ (−2/3, −1/2), outside the earlier range [−1/2, 0). Types (i) and (ii) are identified as Warren-type solutions.
What the classifications cover
The authors interpret the classification in two ways: as a geometric characterization of earlier explicit counterexamples and as a source of a new non-quadratic Euclidean-complete class. That interpretation links the conditional surface classification to the broader existence result, while keeping each conclusion tied to the assumptions stated for the relevant theorem.
The scope is clearest in the classification results. The three-type theorem applies under specified two-dimensional conditions, including the identity condition, A_TT ∥ T and nonconstant |T|, while the flat classification is stated as open parts. The findings concern the defined geometric families and equivalence conditions in the preprint.
The document is an arXiv version-1 preprint dated 26 Aug 2026. The authors report support from the National Natural Science Foundation of China, the Natural Science Foundation of Henan Province, the Postdoctoral Fellowship Program of CPSF and the China Postdoctoral Science Foundation, and they declare no conflict of interest.
Paper data and sources
Original title: New non-quadratic Euclidean complete affine maximal type hypersurfaces via Calabi affine geometry
Authors: Yalin Sun, Cheng Xing, Ruiwei Xu
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text