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Geometry And Topology

# Baecklund-Transformationen und Integrabilitaetsbedingungen by Kirschnick R.

By Kirschnick R.

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Extra info for Baecklund-Transformationen und Integrabilitaetsbedingungen

Example text

2. The third inequality in the Lemma follows from the second one, since 2 W , DX BJ\I (Y, Z) 2 2 2 = DX,Y gJ\I (Z, W ) + DX,Z gJ\I (Y, W ) − DX,W gJ\I (Y, Z) . 6. Lemma. 9 there exist constants c8 , c9 , ˆ such that on any domain UI ⊂ Hn the and c10 depending just on n, h, d0 , and N following estimates hold for any i, i1 , i2 ∈ I with i1 = i2 1/2 (i) Pi GJ\I (1l − Pi ) (ii) Pi BJ\I (1l − Pi )g. , (1l − Pi ) . t. g0 . ´ ` 1 SEMINAIRES & CONGRES , . 41 ANALYTIC MANIFOLDS OF NONPOSITIVE CURVATURE Proof.

ABRESCH V. SCHROEDER converges to p∞ in Hn . 9 ), it is easy to compute the asymptotic behaviour of g ∧ g and T in the limit where pµ → p∞ . 16) (g ∧ g)(vj0 , ξj0 ; ξj0 , vj0 )|pµ ∼ η 2 xj0 (pµ )−1 and ∼ Tj0 (vj0 , ξj0 ; ξj0 , vj0 )|pµ T (vj0 , ξj0 ; ξj0 , vj0 )|pµ ∼ η 2 xj0 (pµ )−2 . 15) implies that R# (vj0 , ξj0 ; ξj0 , vj0 )|pµ is bounded by const η 2 xj0 (pµ )−1 as µ → ∞. Hence, the leading order term in the expansion of Tj0 must cancel versus a suitable counterterm in B ∧ G−1 B. In order to get reasonable estimates for R# on a term by term basis, we need to perform this cancelation explicitly.

In fact, the estimates for LI and BJ\I are proportional to 1/2 1/2 η 2 . However, because of the factors xi1 xi2 (1 +xi1 ) xi ) −1/2 the straightforward bounds for BI −1/2 ∧ G−1 −(1l+GI )−1 (1 +xi2 ) −1/2 BI and BJ\I 1/2 and xi (1 + ∧ G−1 BI are still singular near the divisor. 6 ) actually contain a hidden zero of the same order, and so they only make a contribution to BI ∧ G−1 −(1l+G )−1 I sorbed into −g0 ∧ BI resp. BJ\I ∧ G−1 BI which is small enough to be ab- g0 ; (2) the remaining terms are of such a special nature that their contribution can still be dominated by ε(g0 ∧ g0 + ΦI ) despite the fact that it is singular.