On a convex level set of a plurisubharmonic function and the support of the Monge–Ampère current
Volume 121 / 2018
Annales Polonici Mathematici 121 (2018), 251-262
MSC: Primary 32U15; Secondary 32U35.
DOI: 10.4064/ap180423-14-8
Published online: 12 October 2018
Abstract
We study a geometric property of a continuous plurisubharmonic function which is a solution of the Monge–Ampère equation and has a convex level set. To prove our main theorem, we show a minimum principle for a maximal plurisubharmonic function. By using our results and Lempert’s results, we show a relation between the supports of the Monge–Ampère currents and complex $k$-extreme points of closed balls for the Kobayashi distance in a bounded convex domain in $\mathbb {C}^{n}$.
