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.