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By Lie Sophus, Scheffers G.

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Proof. 2 ) shows that X , KjC |z KjC |z , Y 2 = z , e2j C − z , e1j C C e1j , X z , e2j e1j , Y C + z , e1j e1j , X C e2j , Y C C 2 e2j , X C e2j , Y C + e2j , X C e1j , Y C X , e2j C C C . 2 we have eµj , z 2 C C 2 2 z , ej C X , e2j 2 C z , e1j +2 √ 2 + 1 + 2a(a + 2) eµj , z0 √ 1 ≤ 1 + 2a(a + 2) 1 + 2xj (z0 ) . 6) yield eµj , X 2 C ≤ 1 + 2 eµj , z0 2 X ,X ≤ 1 + 2xj (z0 ) X , X h . 8). ´ ` 1 SEMINAIRES & CONGRES 2 C h C 2 . ANALYTIC MANIFOLDS OF NONPOSITIVE CURVATURE 23 Now, we collect the information.

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. Hence, for these values of η all closed geodesics of the torus are absolutely minimizing elements in their homotopy classes in WIU × (R/2πZ) metric πI∗ g0 + gI #I . In order to pass from the partial to πI∗ (g), we add a positive semidefinite term which vanishes on the torus.

1 (i). ´ E ´ MATHEMATIQUE ´ SOCIET DE FRANCE 1996 44 U. ABRESCH V. SCHROEDER Proof. 7 (ii) that β(η, xi1 ) β(η, xi2 ) ξi1 , LI ξi2 +η 2 h (xi1 ) β(η, xi2 ) vi1 , LI ξi2 + η 2 β(η, xi1 ) h (xi2 ) ξi1 , LI vi2 + η 4 h (xi1 ) h (xi2 ) vi1 , LI vi2 1/2 1/2 ≤ η 2 (1 + 2c4 η 2 )2 cˆ12 (η)xi1 xi2 (1 + xi1 ) −1/2 (1 + xi2 ) −1/2 hence the claim. 13) ∧ LI ∧ pξi vanish identically and that pbi ˆi = B ∧ pbi = −pi 1 + x−1 β(η, xi )2 ξi , LI ξi pi i ∧ ∧ pi pi . 2 (i), we can finish the proof setting cˆ14 := c0 (1 + c4 η)(1 + c4 η 2 ) and cˆ15 := c0 c4 (1 + η + c4 η)(1 + c4 η 2 ) .

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