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)

PDF Abstract
No code implementations yet. Submit your code now