A mass for asymptotically complex hyperbolic manifolds

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hal-00429306, version 1 - 2 Nov 2009 16 A MASS FOR ASYMPTOTICALLY COMPLEX HYPERBOLIC MANIFOLDS. which expands into that is + i 4 + i 4 + i 4 + i 4 1 2 2m j=1 2m j=1 2m j=1 1 2 2m [X·,ej·](∇ g − ∇g0 Aej Aej )φ,φ [X·,ej·] cg(Aej − iJAej)π Ω l−1 − cg0 (Aej − iJAej)π Ω0 l−1 φ,φ 2m j=1 j=1 2m j=1 ([X·,ej·] cg(Aej + iJAej)π Ω l − cg0 (Aej + iJAej)π Ω0 l [X·,ej·](∇ g − ∇g0 Aej Aej )φ,φ [X·,ej·] ([X·,ej·] (ej − iA −1 JAej) · π Ω l−1 − (Aej − iJAej) · π Ω0 l−1 (ej + iA −1 JAej) · π Ω l − (Aej + iJAej) · π Ω0 l In view of lemma 3.4 and corollary 3.5, we are left with : ζφ,φ(AX) ≈ + i 4 + i 4 1 2 2m j=1 2m j=1 2m j=1 [X·,ej·](∇ g − ∇g0 Aej Aej )φ,φ φ,φ , φ,φ φ,φ . [X·,ej·]((ej − iJej) − (Aej − iJAej)) · π Ω0 l−1φ,φ [X·,ej·]((ej + iAej) − (Aej + iJAej)) · π Ω0 l φ,φ . With A = 1 + H we can there there<strong>for</strong>e write ζφ,φ(AX) ≈ I + II + III, with : I := 1 2 II := − i 4 III := − 1 4 2m j=1 2m j=1 2m j=1 [X·,ej·](∇ g − ∇g0 Aej Aej )φ,φ ([X·,ej·]Hej · φ,φ) [X·,ej·]JHej · ˜ φ,φ . The computation of the real part of the first term is classical (cf. [CH] or lemma 10 in [Min], <strong>for</strong> instance) : ReI ≈ − 1 4 (dTrg0 g + divg0 g) |φ|2 .
hal-00429306, version 1 - 2 Nov 2009 A MASS FOR ASYMPTOTICALLY COMPLEX HYPERBOLIC MANIFOLDS. 17 The second term is basically computed in [CH]. Indeed, since H is symmetric, we use the identity [X·,ej·] = 2Xj + 2X · ej· to obtain Re II = − i i (HX · φ,φ) − 2 2 2m j,l=1 Hjl (X · ej · el · φ,φ) = − i i (HX · φ,φ) + Tr H (X · φ,φ) . 2 2 In the same way, the third term can be written Lemma 3.7 — Re III = − 1 (JHX · 2 ˜ φ,φ − 1 2 2m j=1 2m j=1 X · ej · JHej · ˜ φ,φ . X · ej · JHej · ˜ φ,φ ≈ −2 JHX · ˜ φ,φ . Proof. Let us set M := JH and Mij := (ei,Mej), so that 2m j=1 X · ej · JHej · ˜ φ,φ = j,k,p MkjXp ep · ej · ek · ˜ φ,φ . Lemma 3.4 ensures JH ≈ HJ. Since H is symmetric and J antisymmetric, we deduce that M is antisymmetric up to a negligible term. In particular, Mkj ≈ 0 when k = j, hence 2m j=1 X · ej · JHej · ˜ φ,φ ≈ j=k p MkjXp ep · ej · ek · ˜ φ,φ . Given three distinct indices j,k,p we consider the expression ep · ej · ek · ˜ φ,φ = (ep · ej · ek · (φl−1 − φl),(φl−1 + φl)) . Property (7) reduces it into ep · ej · ek · ˜ φ,φ = (ep · ej · ek · φl−1,φl) − (ep · ej · ek · φl,φl−1) . and since the indices are distinct, this is imaginary. So Re 2m j=1 ≈ Re j=k ≈ − Re X · ej · JHej · ˜ φ,φ MkjXj j=k ≈ −2Re j=k MkjXj MkjXj ≈ −2Re MX · ˜ φ,φ ≈ −2 MX · ˜ φ,φ . ej · ej · ek · ˜ φ,φ ek · ˜ φ,φ ek · ˜ φ,φ + Re + Re j=k j=k MkjXk MkjXk ek · ej · ek · ˜ φ,φ ej · ˜ φ,φ
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A mass for asymptotically complex hyperbolic manifolds

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