Équations de Monge-Ampère invariantes sur les variétés Riemanniennes compactes

Abstract

Let (Vn, g) be a smooth n-dimensional compact Riemannian manifold (without boundary). Let g′ be a mapping which assigns to the second covariant jet (in the metric g) of any Ck function φ on Vn, k ⩾ 2, a field g′φ twice covariant and symmetric. We take g′ such that there exists φ ∈ Ck(Vn), k ⩾ 2, admissible i.e. for which Trace [(g′φ)−1⋅∂g′φ∂(∇2φ)] is a new metric (everywhere positive definite) (see below and e.g. [3] [4] [5] [6]). Then, given F smooth, one may consider the following non-linear elliptic problem of Monge-Ampère type: find φ ∈ C∞(Vn) admissible, solution of the equation,M(φ)≡(|g′φ|⋅|g|−1)=exp[F(P,∇φ;φ)],where P denotes a generic point of Vn (the admissibility of φ simply means that the symbol of the differential map d[Log M(φ)] is positive definite, hence the ellipticity at φ). In the present article we give existence and uniqueness results for such a Monge-Ampère problem, not in its full generality, but when prescribing on φ→g′φ to be of a form general to be invariant by changes of unknown function of the type φ → ψ, where: ∀P ∈ Vn, ψ(P) ≡ γ[P, φ(P)], γ(P, t) being a function of C∞(Vn × ℝ) such that ∂γ∂t>0.