Hypersurfaces in 𝑅ⁿ whose unit normal has small BMO norm

Abstract

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.