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A sharp Bernstein theorem for affine maximal type surfaces
Abstract
We investigate Bernstein theorem for the affine maximal type equation ∑i,j=1nfij(Da)ij=0, D=det(fij), a≠0, under completeness of the Calabi metric. In dimension two, we prove that every smooth strictly convex solution on a convex domain is a quadratic polynomial whenever a∉[−2/3,−1/3], resolving a conjecture of Sun and Xu [25]. For every n≥2, we also construct explicit non-quadratic Calabi complete solutions on convex domains for all a∈[−n/(n+1),−1/(n+1)]. These examples show that the rigidity range is sharp in dimension two.
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BibTeX
@misc{sun2026sharp,
title = {{A sharp Bernstein theorem for affine maximal type surfaces}},
author = {Yalin Sun and Cheng Xing and Ruiwei Xu},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.05891},
eprint = {2611.05891},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.05891},
}
RIS
TY - UNPB
TI - A sharp Bernstein theorem for affine maximal type surfaces
AU - Yalin Sun
AU - Cheng Xing
AU - Ruiwei Xu
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.05891
UR - https://arxiv.org/abs/2611.05891
AB - We investigate Bernstein theorem for the affine maximal type equation ∑_i,j=1^nf^ij(D^a)_ij=0, D=det(f_ij), a≠0, under completeness of the Calabi metric. In dimension two, we prove that every smooth strictly convex solution on a convex domain is a quadratic polynomial whenever a∉[−2/3,−1/3], resolving a conjecture of Sun and Xu [25]. For every n≥2, we also construct explicit non-quadratic Calabi complete solutions on convex domains for all a∈[−n/(n+1),−1/(n+1)]. These examples show that the rigidity range is sharp in dimension two.
ER -