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By Malchiodi A.

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During this quantity, that is devoted to H. Seifert, are papers in response to talks given on the Isle of Thorns convention on low dimensional topology held in 1982.

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Non Linaire 17 (2000), no. 1, 47–82. , Perturbation theory for linear operators. Second edition. Grundlehren der Mathematischen Wissenschaften, Band 132. Springer-Verlag, Berlin-New York, 1976. [20] Klingenberg, W. : Riemannian Geometry. Second edition. de Gruyter Studies in Mathematics, 1. , Berlin, 1995. [21] Kwong, M. , Uniqueness of positive solutions of −∆u + u + up = 0 in Rn , Arch. Rational Mech. Anal. 105, (1989), pp. 243-266. [22] Li, Y. , On a singularly perturbed equation with Neumann boundary conditions, Comm.

Springer-Verlag, Berlin-New York, 1976. [20] Klingenberg, W. : Riemannian Geometry. Second edition. de Gruyter Studies in Mathematics, 1. , Berlin, 1995. [21] Kwong, M. , Uniqueness of positive solutions of −∆u + u + up = 0 in Rn , Arch. Rational Mech. Anal. 105, (1989), pp. 243-266. [22] Li, Y. , On a singularly perturbed equation with Neumann boundary conditions, Comm. Partial Differential Equations 23, (1998), pp. 487-545. , The Dirichlet problem for singularly perturbed elliptic equations, Comm.

Here Hζ ⊆ HΣε denotes the eigenspace of TΣε corresponding to ζ and the function u2,ε is defined in Section 3. 1 to the eigenvalues ζ which are close to 0. 9, and let ζ ∈ −δ1 ε2 , δ1 ε2 be an eigenvalue of TΣε . Then there holds 3 ∂ζ 1 = F˜ (σ) + O(ε1− 2 ξ ), ∂ε ε where F˜ (·) is given in (41). Proof. From (65) we obtain 2 (1 − ζ) ε (147) 2 |∇gε u|2 − (1 − ζ) ε Σε |∇g0 u|2 dVg0 ≤ Cε−γ u Sε 2 HΣε . Moreover, reasoning as in the last section of [27], one finds (148) Σε uvup−2 k,ε ∂uk,ε 1 (ε·) = − ∂ε ε uvw0p−2 y · ∇y w0 dVg0 + O(ε−γ ) u Sε HΣε v HΣε for all u, v ∈ HΣε .

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