PreprintCC BY 4.0arXiv
Gradient estimates and Liouville theorems for exponentially harmonic functions
Abstract
We study exponentially harmonic functions on complete Riemannian manifolds. Under a lower Ricci curvature bound Ricg≥−Kg, we first establish a local gradient estimate in terms of the oscillation. We then prove a pointwise gradient estimate for solutions bounded from one side. As an application, when Ricg≥0, we obtain a Liouville theorem under a one-sided sublinear growth condition. In particular, every exponentially harmonic function bounded above or below is constant.
Related
- Gradient estimates and Liouville-type theorems for the semilinear elliptic equation involving the nonlinear gradient source
- Liouville-type theorems and universal estimates for semilinear elliptic equations involving the product of the function and its gradient
- Hamilton's gradient estimates and Liouville theorems for ut=Δum+aulog u+bu on Riemannian manifolds
- Solutions of the space-fractional heat equation with polynomial growth: Liouville theorems and Schauder estimates
- Liouville Theorems for Semilinear Parabolic Equations Involving the Nonlinear Gradient Term
BibTeX
@misc{jin2026gradient,
title = {{Gradient estimates and Liouville theorems for exponentially harmonic functions}},
author = {Xishen Jin and Xi Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.35571},
eprint = {2610.35571},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.35571},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Gradient estimates and Liouville theorems for exponentially harmonic functions
AU - Xishen Jin
AU - Xi Zhang
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.35571
UR - https://arxiv.org/abs/2610.35571
AB - We study exponentially harmonic functions on complete Riemannian manifolds. Under a lower Ricci curvature bound Ric_g≥−Kg, we first establish a local gradient estimate in terms of the oscillation. We then prove a pointwise gradient estimate for solutions bounded from one side. As an application, when Ric_g≥0, we obtain a Liouville theorem under a one-sided sublinear growth condition. In particular, every exponentially harmonic function bounded above or below is constant.
ER -