Monge-Ampère equations relative to a Riemannian metric

Abstract:

We prove that in a bounded strictly convex open set $\Omega$ in $\mathbb {R}^n$, the problem \[ \begin {cases} \det \nabla ^2u=f(x),\ u|_{\partial \Omega }=\varphi , \end {cases}\] where $f>0,f\in C^\infty (\overline \Omega ), \varphi \in C^\infty (\partial \Omega )$, has a unique strictly convex solution $u\in C^\infty (\overline \Omega )$. This result extends to an arbitrary metric a theorem which has been proved by Caffarelli-Nirenberg-Spruck in the case of the Euclidean metric.