Interior gradient estimates for mean curvature equations on Riemannian manifolds

Weiyang Li; Fanheng Xu

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arXiv · 2610.24745

Interior gradient estimates for mean curvature equations on Riemannian manifolds

Weiyang Li·Fanheng Xu

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Abstract

In this paper, we study interior gradient estimates for solutions of \[ \operatorname{div}_{M}\left( \frac{\nabla u}{\sqrt{1+|\nabla u|_g^2}} \right) =f(x,\nabla u) \] on a complete Riemannian manifold of dimension $n\ge2$. 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.

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BibTeX RIS

Weiyang Li, Fanheng Xu. 2026-10-22. Interior gradient estimates for mean curvature equations on Riemannian manifolds. https://arxiv.org/abs/2610.24745

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