Let \(K\) be a compact subset of \(\mathbb C^2\). Let \[ L(u):= (\delta_{\alpha\beta}-|\partial u|^{-2}u_{\bar{\alpha}}u_{\beta})u_{\alpha \bar{\beta}} \] denote the Levi operator, where \(u_{\alpha}=\partial u/\partial z_{\alpha},\quad \alpha =1,2\), and \((z_1,z_2)\in\mathbb C^2\). In their earlier papers (Zbl 0888.32007 and Zbl 1009.32008) the authors proved that if \(K\) is a zero set of a continuous function \(g\), then the parabolic problem \[ \begin{alignedat}{2} u_t &= L(u) &\quad &{\text{in}} \quad \mathbb C^2\times(0,+\infty)\\u &=g &\quad &{\text{on}}\quad \mathbb C^2\times \{0\}\end{alignedat} \] has a unique weak (viscosity) solution \(u=u(x,t),\quad x=(z_1,z_2)\), which is constant for \(|x|+t \gg 0\) and that the family \(\{\mathcal E_t^{\mathcal L}(K)\}_{t\geq 0}\) with \(\mathcal E_t^{\mathcal L}(K)= \{x\in \mathbb C^2:u(x,t)=0\}\) is independent of \(g\). The family is called the evolution of \(K\).
Main conclusion of the paper: Let \(\Omega \subset \mathbb C^2\) be a bounded convex domain. Then the evolution of \(b\Omega\) is strictly contracting (i.e., \(\Gamma_t=\mathcal E_t^{\mathcal L}(b{\Omega})\subset \Omega\) for all \(t> 0\)) and is of stationary type, i.e., \(\Gamma_{t_1}\cap \Gamma_{t_2} = \emptyset, \quad t_1\neq t_2\).