Evolution by Levi form in ${\mathbb{P}}^2$

Abstract

In this paper we introduce the notion of evolution by Levi form of a closed set $K$ in the complex projective space ${\mathbb{P}}^2$, which is governed by a parabolic problem for the Levi operator. The main result of the paper is the following: if $K$ is the closure of a pseudoconvex domain with a regular boundary then the evolution is contained in $K$. As a consequence, a hypothetic Levi flat hypersurface in ${\mathbb{P}}^2$ would be left fixed by evolution.