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!



Bernstein type theorems for CMC hypersurfaces in $\mathbb{R}^{n+1}$
This was solved in a series of articles in the 1960s.


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.


Here is a link to the latter article:


*Bombieri, E.; De Giorgi, E.; Giusti, E.*, [**Minimal cones and the Bernstein problem**](http://dx.doi.org/10.1007/BF01404309), Invent. Math. 7, 243-268 (1969). [ZBL0183.25901](https://zbmath.org/?q=an:0183.25901).


