Translating solutions of the nonparametric mean curvature flow with nonzero Neumann boundary data in product manifold $M^n\times \mathbb{R}$

In this paper, we can prove the existence of translating solutions to the nonparametric mean curvature flow with nonzero Neumann boundary data in a prescribed product manifold $M^n\times \mathbb{R}$, where $M^n$ is an $n$-dimensional ($n\ge 2$) complete Riemannian manifold with nonnegative Ricci curvature, and $\mathbb{R}$ is the Euclidean $1$-space.