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Rigidity of Convex Entire Solutions to Sum Hessian Equations
Abstract
We study the rigidity of convex entire solutions to sum Hessian equations. Using a dimension-lifting identity, we reduce the problem to a standard k-Hessian equation in one higher dimension. By applying existing rigidity theorems, we prove that every C2 convex entire solution is a quadratic polynomial for 2≤ k≤ n, α≥0, and a positive constant right-hand side, without any growth assumption at infinity. We also discuss the linear case k=1.
Related
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BibTeX
@misc{ren2026rigidity,
title = {{Rigidity of Convex Entire Solutions to Sum Hessian Equations}},
author = {Changyu Ren and Xianlu Sun},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.33397},
eprint = {2610.33397},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.33397},
}
RIS
TY - UNPB
TI - Rigidity of Convex Entire Solutions to Sum Hessian Equations
AU - Changyu Ren
AU - Xianlu Sun
PY - 2026
DA - 2026/10/24
PB - arXiv
DO - 10.48550/arxiv.2610.33397
UR - https://arxiv.org/abs/2610.33397
AB - We study the rigidity of convex entire solutions to sum Hessian equations. Using a dimension-lifting identity, we reduce the problem to a standard k-Hessian equation in one higher dimension. By applying existing rigidity theorems, we prove that every C^2 convex entire solution is a quadratic polynomial for 2≤ k≤ n, α≥0, and a positive constant right-hand side, without any growth assumption at infinity. We also discuss the linear case k=1.
ER -