I am interested in the following form of isoperimetric inequality for Kähler Manifolds (for example unit ball $B^n\subset \mathbb{C}^n$ with Bergman metric). It should say something like this: if $F$ is a smooth domain in the Bergman ball with the Bergman volume $V$, whose boundary $S$ as a submanifold of Bergman ball has area $A$, then I expect that this is true $V \le V_r$ where $V_r$ is the volume of the ball centered at $0$ having area equal to $A$. Is there any reference for this? It is well-known that the Bergman metric has negative sectional curvature and perhaps this can help?
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