Équations du type de Monge—Ampère sur les variétés Riemanniennes compactes, II

AbstractLet (Vn, g) be a C∞ compact Riemannian manifold. For suitable functions ϑ, we consider the change of metric: g′ = g + Hess(ϑ), and the associated Monge-Ampère equation Log(¦g′¦ · ¦g¦−1) = −λϑ + ƒ, ƒ ϵ C∞(Vn), in the open case λ ⩾ 0, where the equation is not a priori locally invertible. We study first the case λ = 0 by turning the original equation into a “close enough” locally invertible one, namely, Log(¦g′¦ · ¦g¦−1) = ƒ −∝ ϑ dV, which is solved in C∞ by the continuity method. Then, discarding local invertibility, we solve in C∞ the case λ > 0 by mean of a fixed point argument. Moreover the study of the case λ = 0 leads to corollaries about equations such as Log(¦g′¦ · ¦g¦−1) = F(P, ▽ϑ; ϑ) − a(F) ∝ ϑ dV, a(F) a suitable constant depending only on F and (Vn g), with particularly weak assumptions on the function F