Rearrangements and the Monge–Ampére equations

Comparison results for rearrangements of solutions to elliptic equations are known for linear equations (Talenti) and for the real Monge–Ampére equation (Talenti, Tso). The main tool in the proof is always an appropriate isoperimetric inequality. These results often lead to optimal estimates, as they reduce the problem to radially symmetric solutions. We will discuss a similar question for the complex Monge–Ampére equation.