Interior gradient estimates for mean curvature equations on Riemannian manifolds

In this paper, we study interior gradient estimates for solutions of $${\mathrm{div}}_M\left(\frac{\mathrm{\nabla }u}{\sqrt{1+\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}_g^2}}\right)=f(x,\mathrm{\nabla }u)$$ on a complete Riemannian manifold of dimension $n\ge 2$. 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 explicitly separate the contributions of curvature, distance comparison, and the dependence of $f$ on the base point and the tangent vector. As applications, we obtain Liouville-type results for entire solutions under suitable curvature and growth assumptions.