Some totally real embeddings of three-manifolds

Abstract

LetV be a complex hypersurface in an open subset of ℂ3, and letM be a smooth compact real hypersurface inV. Using a theorem of Gromov we prove that there exist small C1 perturbations\(\tilde M\) ofM in ℂ3 such that\(\tilde M\) is a totally real submanifold of ℂ3. As a consequence we show that certain quotients of the three-sphere admit totally real embeddings into ℂ3. In some special cases including the real projective three-space we find explicit totally real embeddings into ℂ3. Our construction is similar to that of Ahern and Rudin who found a totally real embedding of the three-sphere into ℂ3.