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**Sample text**

11) 2 DX,W g(Y, Z) −2 h (xj ) − 4x−1 4η 2 x−2 j j h (xj ) + 6xj h(xj ) = j∈J Kj , DX Kj Kj , DW Kj Kj , Y Kj , Z 2η 2 x−2 h (xj )−2x−1 j j h(xj ) + j∈J Kj , Y Kj , Z DX Kj , DW Kj − R0 (Kj , X)W , Kj + Kj , Y Kj , DW Kj DX Kj , Z + Kj , DX Kj DW Kj , Z + Kj , Z Kj , DX Kj DW Kj , Y + Kj , DW Kj DX Kj , Y η 2 x−2 j h(xj ) + j∈J DW Kj , Y DX Kj , Z + DX Kj , Y − R0 (Kj , X)W , Y Kj , Z − Kj , Y DW Kj , Z R0 (Kj , X)W , Z . 2 In this computation we have used the identity DX,W Kj + R0 (Kj , X)W = 0 in order to determine the second derivatives D2 Kj .

Remark. 10 that supp(h) ⊂ [0, sinh2 1 2 d0 ]. 18 ) − 2η 2 h (xi ) g0 i∈I − ∧ pξi − 2η 2 (1 + xi ) h (xi ) pi ∧ pi i∈I 4 η h (xi1 ) h (xi2 ) pξi1 ∧ pξi2 . 8 at all. 4. ´ E ´ MATHEMATIQUE ´ SOCIET DE FRANCE 1996 6. 5. 1. Proposition. 9. Then, ˆ ε such that for 0 < η < η2 for any ε > 0, there exists a constant η2 = η2 n, h, d0 , N, the following estimates hold on each domain Ω ∩ UI (i) −ε g0 (ii) −ε g0 ∧ ∧ g 0 + ΦI g 0 + ΦI ≤ ≤ BI ∧ G−1 −(1l+GI )−1 BI BJ\I ∧ G−1 BI ≤ ε g 0 ∧ g 0 + ΦI ≤ ε g 0 ∧ g 0 + ΦI .

We get πI∗ (g0 + gI )|(p,ϕ) = g0 |Tp W U ×Tp W U + I xi + η 2 h(xi ) I |p dϕi 2 . i∈I Evidently, the right hand side describes a real analytic, Stab I -invariant, Riemannian #I metric on all of WIU × (R/2πZ) . (iii) Note that p ∈ SI is contained in some domain UI with I ⊂ I ⊂ J. 5) it is clear that πI−1 {p} is a totally geodesic product torus in WIU × (R/2πZ) #I equipped with the metric πI∗ g0 + gI . If η is suﬃciently small, then the function x → x + η 2 h(x), x ≥ 0, takes its absolute minimum precisely at x = 0.