Some results on evolution

TL;DR

This paper studies the evolution of compact subsets in complex space under Levi form-driven dynamics, proving existence, uniqueness, and long-term convergence to Levi flat hypersurfaces.

Contribution

It establishes the existence and uniqueness of solutions to a Levi form evolution problem and describes the asymptotic behavior of evolving graphs.

Findings

Existence and uniqueness of the evolution solution for all time.

Evolution of smooth graphs remains a graph over time.

Long-term convergence to Levi flat hypersurfaces when boundary conditions are met.

Abstract

Let $K$ be a compact subset of $\mathbb C^n$, $K^\ast K$ a closed subset. In this paper we are dealing with evolution $E_t(K,K^\ast)$ of $K$ with fixed part $K^\ast$ by Levi form. This amounts to solve a parabolic problem for an elliptic operator. We prove existence and unicity for such a problem and the solution $u(z,t)$ exists for any time $t\ge 0$.If $K$ is a smooth graph $\Gamma$ and $K^\ast={\rm b}\Gamma$ the the evolution $E_t(\Gamma,{\rm b}\Gamma)$ is still a graph. In particular, if ${\rm b}\Gamma$ bounds a Levi flat hypersurface $M$ then $E_t(\Gamma,{\rm b}\Gamma)\to M$ as $t\to+\infty$.