Gradient Estimates of Mean Curvature Equation with Neumann Boundary Condition in $ M^n\times \mathbb{R}$

Abstract: In this note, we study the prescribed mean curvature equation with Neumann boundary conditions on Riemannian product manifold $ M^n\times \mathbb{R}$. The main goal is to establish the boundary gradient estimates for solutions by the maximum principle. As a consequence, we obtain an existence result.