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A growth gap for anisotropic minimal graphs
Abstract
We prove a growth gap for the gradient of entire anisotropic minimal graphs. For each n≥2 and each smooth uniformly elliptic parametric integrand Φ on ℝn+1, there is an exponent α=α(n,Φ)>0 such that every smooth nonaffine entire Φ-minimal graph u:ℝn→ℝ satisfies supBRn|Du|≥ cRα for all R≥ R0, for some constants c>0 and R0<∞ depending on the solution. In particular, gradient growth o(Rα) forces flatness, resolving a conjecture of Mooney and the author. The result holds in every dimension, without any assumption that Φ is close to the Euclidean area integrand.
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BibTeX
@misc{yang2026growth,
title = {{A growth gap for anisotropic minimal graphs}},
author = {Yang Yang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.08972},
eprint = {2609.08972},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.08972},
}
RIS
TY - UNPB
TI - A growth gap for anisotropic minimal graphs
AU - Yang Yang
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2609.08972
UR - https://arxiv.org/abs/2609.08972
AB - We prove a growth gap for the gradient of entire anisotropic minimal graphs. For each n≥2 and each smooth uniformly elliptic parametric integrand Φ on ℝ^n+1, there is an exponent α=α(n,Φ)>0 such that every smooth nonaffine entire Φ-minimal graph u:ℝ^n→ℝ satisfies sup_B_R^n|Du|≥ cR^α for all R≥ R_0, for some constants c>0 and R_0<∞ depending on the solution. In particular, gradient growth o(R^α) forces flatness, resolving a conjecture of Mooney and the author. The result holds in every dimension, without any assumption that Φ is close to the Euclidean area integrand.
ER -