Lewy’s curves and chains on real hypersurfaces

Abstract:

Lewy’s curves on an analytic real hypersurface $M = \{ r(z,z) = 0\}$ in ${{\mathbf {C}}^2}$ are the intersections of $M$ with any of the Segre hypersurfaces ${Q_w} = \{ z:r(z,w) = 0\}$. If $M$ is the standard unit sphere, these curves are chains in the sense of Chern and Moser. This paper shows the converse in the strictly pseudoconvex case: If all of Lewy’s curves are chains, $M$ is locally biholomorphically equivalent to the sphere. This is proven by analyzing the holomorphic structure of the space of chains. A similar statement is true about real hypersurfaces in ${{\mathbf {C}}^n}$, $n > 2$, in which case the proof relies on a pseudoconformal analogue to the theorem in Riemannian geometry which states that a manifold having "sufficiently many" totally geodesic submanifolds is projectively flat.