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We study that the n-graph defined by a smooth map $${f:\Omega\subset\mathbb R^{n}\to \mathbb R^{m}, m\ge 2,}$$ in $${\mathbb R^{m+n}}$$ of the prescribed mean curvature and the Gauss image. Under the condition     $$\Delta_f=\left[\text{det}\left(\delta_{ij}+\sum_\alpha{\frac {\partial {f^\alpha}}{\partial {x^i}}}{\frac {\partial {f^\alpha}}{\partial {x^j}}}\right)\right]^{\frac{1}{2}}      we derive the interior curvature estimates     $$\sup_{D_R(x)}|B|^2\le{\frac{C}{R^2}}$$     when 2 ≤ n ≤ 5 with constant C depending on the given geometric data. If there is no dimension limitation we obtain     $$\sup_{D_R(x)}|B|^2\le CR^{-a}\sup_{D_{2R}(x)}(2-\Delta_f)^{-\left({\frac{3}{2}}+{\frac{1}{s}}\right)},\quad s=\min(m, n)$$     with a $${{\frac{\sqrt{2}}{4}}\pi}$$ in G  n,m  we also derive corresponding estimates.

Curvature estimates for submanifolds with prescribed Gauss image and mean curvature