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arXiv · 2410.15202

Asympotitcs for Some Singular Monge-Ampère Equations

Nicholas McCleerey
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Abstract

Given a psh function $φ\in\mathcal{E}(Ω)$ and a smooth, bounded $θ\geq 0$, it is known that one can solve the Monge-Ampère equation $\mathrm{MA}(φ_θ)=θ^n\mathrm{MA}(φ)$, with some form of Dirichlet boundary values, by work of Ahag--Cegrell--Czyż--Hiep. Under some natural conditions, we show that $φ_θ$ is comparable to $θφ$ on much of $Ω$; especially, it is bounded on the interior of $\{θ= 0\}$. Our results also apply to complex Hessian equations, and can be used to produce interesting Green's functions.

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Nicholas McCleerey. 2024-10-19. Asympotitcs for Some Singular Monge-Ampère Equations. https://arxiv.org/abs/2410.15202

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