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Let      M M       be a hypersurface in      R d + 1 {{\mathbf {R}}^{d + 1}}       whose Gauss map has small BMO norm. This condition is closely related to (but much weaker than) the requirement that the principal curvatures of      M M       have small      L d ( M ) {L^d}\left ( M \right )       norm. (The relationship between these two conditions is a nonlinear geometrical analogue of a classical Sobolev embedding.) This paper deals with the problem of understanding the geometrical constraints imposed on      M M       by the requirement that the Gauss map have small BMO norm.

Hypersurfaces in ${\bf R}\sp n$ whose unit normal has small BMO norm