Équations du type Monge-Ampère sur les variétés Riemanniennes compactes, I

AbstractLet (Vn, g) be a C∞ compact Riemannian manifold. For a suitable function ϑ on Vn, let us consider the change of metric: g′ = g + Hess(ϑ), and the function, as a ratio of two determinants, M(ϑ) = ¦g′¦ ¦g¦−1. Using the method of continuity, we first solve in C∞ the problem: Log M(ϑ) = λϑ + ƒ, λ > 0, ƒ ϵ C∞. Then, under weak hypothesis on F, we solve the general equation: Log M(ϑ) = F(P, ϑ), F in C∞(Vn × ¦α, β¦), using a method of iteration. Our study gives rise to an interesting a priori estimate on ¦▽ϑ¦, which does not occur in the complex case. This estimate should enable us to solve the equation above when λ ⩽ 0, providing we can overcome difficulties related to the invertibility of the linearised operator. This open question will be treated in our next article