Équations de Monge-Ampère invariantes sur les variétés Riemanniennes compactes. (French) Zbl 0555.58026

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.
Reviewer: N.Jacob


58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)
35J60 Nonlinear elliptic equations
35Q99 Partial differential equations of mathematical physics and other areas of application
58J70 Invariance and symmetry properties for PDEs on manifolds
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