# Dimension of the minimum set for the real and complex Monge-Amp\{e}re equations in critical Sobolev spaces

15 Mar 2017 Collins Tristan C. Mooney Connor

We prove that the zero set of a nonnegative plurisubharmonic function that solves $\det (\partial \overline{\partial} u) \geq 1$ in $\mathbb{C}^n$ and is in $W^{2, \frac{n(n-k)}{k}}$ contains no analytic sub-variety of dimension $k$ or larger. Along the way we prove an analogous result for the real Monge-Amp\ere equation, which is also new... (read more)

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