Conformal scalar curvature on the hyperbolic space

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Conformal scalar curvature on the hyperbolic space Erwann DELAY? University of Nice-Sophia Antipolis Abstract. Regarding the theme of the conformal scalar curvature on the hyperbolic space, we bring here a study of the fine asymptotic behavior in any dimension. We always deal with general semi-linear equations, before applying our results to the particu- lar case of the geometric equation. 1 Preliminary Let B the unit ball in Rn (with the euclidien metric E) and let ? the function definite on B by: ?(x) = 1 2 (1? |x|2). The model of hyperbolic spaceHn(?1) chosen isB with the conformal metric : H0 = ? ?2E. For v > ?1 real function on B, we define the conformal metric Hv = (1 + v)H0. We are interested in the scalar curvature of such conformal metrics. We are going to show that the map v ?? Scal(Hv)? Scal(H0) is invertible near zero in some appropriate spaces. Let u in Ck,?loc (B), we will say that u is in ?sk,? if the next quantity, who represents his norm on this space, is finite : ? u ?sk,?:= ∑ |?|≤k sup x?B [?(x)?s+|?||∂?u(x)|] + ∑ |?|=k sup x,y?B x 6=y min(?

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Conformal scalar curvature on the hyperbolic space

∗ Erwann DELAY

University of Nice-Sophia Antipolis

Abstract. Regarding the theme of the conformal scalar curvature on the hyperbolic space, we bring here a study of the ﬁne asymptotic behavior in any dimension.We always deal with general semi-linear equations, before applying our results to the particu-lar case of the geometric equation.

1 Preliminary n LetBthe unit ball inR(with the euclidien metricE) and letρthe function deﬁnite onBby: 1 2 ρ(x(1) =− |x|). 2 n The model of hyperbolic spaceH(−1) chosen isBwith the conformal metric : −2 H0=ρ E. Forv >−1 real function onB, we deﬁne theconformal metric Hv= (1 +v)H0. We are interested in thescalar curvatureof such conformal metrics.We are going to show that the map v−→Scal(Hv)−Scal(H0) k,α is invertible near zero in some appropriate spaces.LetuinC(B), we will loc s say thatuis in Λif the next quantity, who represents his norm on this k,α space, isﬁnite: X s−s+|γ|γ kuk:= sup[ρ(x)|∂ u(x)|] k,α x∈B |γ|≤k X γ γ |∂ u(x)−∂ u(y)| −s+k+α−s+k+α + supmin(ρ(x), ρ(y) ). α x,y∈B|x−y| |γ|=k x6=y

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