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

## TL;DR

This paper investigates the structure of zero sets for solutions to real and complex Monge-Ampère equations within critical Sobolev spaces, establishing sharp bounds on the dimension of analytic sub-varieties they can contain.

## Contribution

It proves new sharp bounds on the dimension of zero sets for Monge-Ampère solutions in critical Sobolev spaces, including an extension of interior regularity results.

## Key findings

- Zero sets contain no analytic sub-variety of dimension k or larger.
- Results are sharp, matching known examples.
- Extension of interior regularity to critical Sobolev spaces.

## Abstract

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. These results are sharp in view of well-known examples of Pogorelov and B{\l}ocki. As an application, in the real case we extend interior regularity results to the case that $u$ lies in a critical Sobolev space (or more generally, certain Sobolev-Orlicz spaces).