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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!

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1 Answer 1

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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, Invent. Math. 7, 243-268 (1969). ZBL0183.25901.

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  • $\begingroup$ 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 $n \ge 9$? $\endgroup$
    – Onil90
    Nov 19, 2018 at 11:54
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    $\begingroup$ 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. $\endgroup$ Nov 19, 2018 at 11:59

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