The well-known Half-space Theorem by Hoffman and Meeks says that there is no nonflat complete properly embedded minimal surface contained in an half space of $\mathbb{R}^3$.

The higher dimensional version of this statement is false. Indeed the higher dimensional cathenoid is contained in a slab between two parallel hyperplanes.

My question is if there is any known example of a complete minimal hypersurface $\Sigma^n \subset \mathbb{R}^{n+1}$, with $n >2$, such that

  1. $\Sigma$ is contained in a half-space of $\mathbb{R}^{n+1}$;

  2. $\Sigma$ is not contained in any slab.

Probably it is possible to build something like that maybe gluing together an infinite family of hyperplanes (contained in a half-space) with cathenoidal necks, but I don't know if it has been done.

Any help will be very much appreciated!! Thanks!



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