Bernstein-type theorem of translating solitons in arbitrary codimension with flat normal bundle

We give a proof for a Bernstein-type theorem of complete graphs of translating solitons in higher codimension. In the case of hypersurfaces, Bao–Shi showed that a translating soliton whose image of the Gauss map is contained in a compact subset in an open hemisphere is a hyperplane. This means that there is no nontrivial translating soliton whose slope is bounded. In the present article, we generalize this theorem in arbitrary codimension. Moreover we obtain an optimal growth condition which allows unbounded slopes. As a corollary, our result covers a classical Bernstein-type theorem for minimal submanifolds.