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In this paper, we prove a monotonicity formula and some Bernstein type results for translating solitons of hypersurfaces in $$\mathbb {R}^{n+1}$$ R n + 1 , giving some conditions under which a translating soliton is a hyperplane. We also show a gap theorem for the translating soliton of hypersurfaces in $$R^{n+k}$$ R n + k , namely, if the $$L^n$$ L n norm of the second fundamental form of the soliton is small enough, then it is a hyperplane.

Bernstein theorem for translating solitons of hypersurfaces