Équations de Monge-Ampère invariantes sur les variétés Riemanniennes compactes (Q760699)

Let \((V_ n,g)\) be a smooth n-dimensional compact Riemannian manifold without boundary. Let \(g'\) be a map which assigns to the second covariant jet (in the metric \(g\)) of any \(C^ k\) function \(\phi\) on \(V_ n\), \(k\geq 2\), a field \(g'_{\phi}\) twice covariant and symmetric. The author takes g' such that there exists \(\phi\in C^ k(V_ n)\), \(k\geq 2\), admissible, i.e. for which \(Tr[(g'_{\phi})^{-1}\cdot \partial g'_{\phi}/\partial (\nabla^ 2\phi)]\) is a new metric. Then, given F smooth, one may consider the following nonlinear elliptic problem of Monge-Ampère type: find \(\phi \in C^{\infty}(V_ n)\) admissible, solution of the equation \[ M(\phi):=(| g'_{\phi}| \cdot | g|^{-1})=\exp [F(P,\nabla \phi;\phi)], \] where P denotes a generic point of \(V_ n\) (the admissibility of \(\phi\) means that the symbol of the differential map \(d(\log M(\phi))\) is positive definite). In the present article the author gives some existence and uniqueness results for such Monge-Ampère problem under additional invariance conditions.