# Bernstein type theorems for CMC hypersurfaces in $\mathbb{R}^{n+1}$

Is there any Bernstein type theorems for CMC hypersurfaces in $$\mathbb{R}^{n+1}$$ in the literature?

More precisely I would like to know if there is an answer to the following

QUESTION: Let $$f : \mathbb{R}^n \to \mathbb{R}$$ be a smooth function such that $$\mathrm{graph}(f)$$ is a constant mean curvature hypersurface of $$\mathbb{R}^{n+1}$$. Is it true that $$\mathrm{graph}(f)$$ must be an affine hyperplane?

I don't know much about CMC hypersurfaces and I don't know where to look for an answer. Even if the question has a negative answer, I would like to know if there are counterexamples or if one can get an affirmative answer under some volume growth condition.

Any help will be very much appreciated! Thanks!

De Giorgi, Almgren, and Simons have shown that in $$\mathbb{R}^{\le 8}$$ every CMC graph is a hyperplane. Then Bombieri - De Giorgi - Giusti have shown that in $$\mathbb{R}^{\ge 9}$$ there are minimal graphs which are not hyperplanes.
• Thanks a lot for the answer! I forgot to say that I am mostly interested in the "non-minimal" case, i.e. when $H \ne 0$. What is it known in that case for dimension $n \ge 9$? Nov 19, 2018 at 11:54
• By Chern'65 and Flanders' 66, if the absolute value of mean curvature of a graph is at least $H$, then the domain of the graph is contained in a ball of radius $1/H$. This implies that the only CMC graphs over $\mathbb{R}^n$ are minimal. Nov 19, 2018 at 11:59