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A Cheng-Yau type estimate for positive biharmonic functions
Abstract
We establish a Cheng-Yau type estimate for positive biharmonic functions on complete Riemannian manifolds with Ricci curvature satisfies Ricg ≥ −(n−1)K g. If u is a positive biharmonic function in B2R(p), then −(Δg u)/u +1/8n(|∇ u|g2)/u2 ≤ Cn(R−2+K) on BR(p). Further, if the Ricci curvature is nonnegative, every global positive biharmonic function satisfies Δg u≡ c and the sharp estimate |∇ u|g2≤2cu for some nonnegative constant c. We also show that every positive k-polyharmonic function has nonnegative constant (k−1)-st Laplacian and growth of order at most 2k−2.
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BibTeX
@misc{jiang2026chengyau,
title = {{A Cheng-Yau type estimate for positive biharmonic functions}},
author = {Guosheng Jiang and Mingxiang Li and Zhehui Wang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.05965},
eprint = {2609.05965},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.05965},
}
RIS
TY - UNPB
TI - A Cheng-Yau type estimate for positive biharmonic functions
AU - Guosheng Jiang
AU - Mingxiang Li
AU - Zhehui Wang
PY - 2026
DA - 2026/09/05
PB - arXiv
DO - 10.48550/arxiv.2609.05965
UR - https://arxiv.org/abs/2609.05965
AB - We establish a Cheng-Yau type estimate for positive biharmonic functions on complete Riemannian manifolds with Ricci curvature satisfies Ric_g ≥ −(n−1)K g. If u is a positive biharmonic function in B_2R(p), then −(Δ_g u)/u +1/8n(|∇ u|_g^2)/u^2 ≤ C_n(R^−2+K) on B_R(p). Further, if the Ricci curvature is nonnegative, every global positive biharmonic function satisfies Δ_g u≡ c and the sharp estimate |∇ u|_g^2≤2cu for some nonnegative constant c. We also show that every positive k-polyharmonic function has nonnegative constant (k−1)-st Laplacian and growth of order at most 2k−2.
ER -