Équations du type Monge-Ampère sur les variétés Riemanniennes compactes, I |
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Authors: | Philippe Delanoë |
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Affiliation: | Département de Mathématiques de l''Université Pierre et Marie Curie, Paris, France |
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Abstract: | Let (Vn, g) be a C∞ compact Riemannian manifold. For a suitable function on Vn, let us consider the change of metric: g′ = g + Hess(), and the function, as a ratio of two determinants, M() = ¦g′¦ ¦g¦−1. Using the method of continuity, we first solve in C∞ the problem: Log M() = λ + ƒ, λ > 0, ƒ ε C∞. Then, under weak hypothesis on F, we solve the general equation: Log M() = F(P, ), F in C∞(Vn × ¦α, β¦), using a method of iteration. Our study gives rise to an interesting a priori estimate on ¦¦, 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. |
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