Real and complex k-planes in convex hypersurfaces

TL;DR

This paper investigates the geometric structure of convex hypersurfaces in real and complex spaces, showing that bounds on the second fundamental form or Levi form imply the existence of certain tangent planes near any point.

Contribution

It establishes a link between the rank bounds of the second fundamental form or Levi form and the local existence of tangent planes of specific dimensions on convex hypersurfaces.

Findings

Bound on the second fundamental form implies existence of real tangent planes.

Bound on the Levi form implies existence of complex tangent planes.

Results apply to smooth convex hypersurfaces in both real and complex spaces.

Abstract

It is shown that that the rank of the second fundamental form (resp. the Levi form) of a $\mathcal C^2$-smooth convex hypersurface $M$ in $\Bbb R^{n+1}$ (resp. $\Bbb C^{n+1}$) does not exceed an integer constant $k<n$ near a point $p\in M,$ then through any point $q\in M$ near $p$ there exists a real (resp. complex) $(n-k)$-dimensional plane that locally lies on $M.$