Interior gradient estimates for mean curvature equations on Riemannian manifolds
Abstract
In this paper, we study interior gradient estimates for solutions of divM( (∇ u)/(√(1+|∇ u|g2)) ) =f(x,∇ u) on a complete Riemannian manifold of dimension n≥2. Under a Ricci curvature lower bound and a Hessian comparison for the distance function, we derive polynomial and exponential gradient bounds under structural conditions on f and its horizontal and vertical derivatives. The estimates quantify the dependence on the radius, the oscillation of the solution, and the geometric and analytic parameters. In the Euclidean case, we also improve the dependence on the radius in the gradient estimate of Xu~\cite[Theorem~1.1(d)]{Xu2026}. As applications, we obtain Liouville-type results for entire solutions under suitable curvature and growth assumptions.
Related
- Interior curvature estimates for graphical curvature quotient equations
- Differential Harnack Estimates and Frequency Monotonicity for Filtration Equations on Riemannian Manifolds
- Hamilton-Type Gradient Estimates for Porous Medium and Fast Diffusion Equations on Riemannian Manifolds for all exponents
- Interior Curvature Estimates for the Graphical Scalar Curvature Equation in All Dimensions
- Interior curvature estimates for convex solutions of a prescribed curvature quotient equation
BibTeX
@misc{li2026interior,
title = {{Interior gradient estimates for mean curvature equations on Riemannian manifolds}},
author = {Weiyang Li and Fanheng Xu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.24745},
eprint = {2610.24745},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.24745},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Interior gradient estimates for mean curvature equations on Riemannian manifolds
AU - Weiyang Li
AU - Fanheng Xu
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.24745
UR - https://arxiv.org/abs/2610.24745
AB - In this paper, we study interior gradient estimates for solutions of div_M( (∇ u)/(√(1+|∇ u|_g^2)) ) =f(x,∇ u) on a complete Riemannian manifold of dimension n≥2. Under a Ricci curvature lower bound and a Hessian comparison for the distance function, we derive polynomial and exponential gradient bounds under structural conditions on f and its horizontal and vertical derivatives. The estimates quantify the dependence on the radius, the oscillation of the solution, and the geometric and analytic parameters. In the Euclidean case, we also improve the dependence on the radius in the gradient estimate of Xu~\cite[Theorem~1.1(d)]{Xu2026}. As applications, we obtain Liouville-type results for entire solutions under suitable curvature and growth assumptions.
ER -