# Modern algebraic topology by D G Bourgin By D G Bourgin

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Example text

C \ Z open subsets such that ν Ων = C \ Z, and for ν ∈ N let wν = (uν , Aν ) ∈ WΩp ν be an Rν -vortex. Assume that there exists a compact subset K ⊆ M such that for ν large enough uν (Ων ) ⊆ K. 36) sup ν eR wν L∞ (Q) ν ∈ N : Q ⊆ Ων < ∞. Then there exists an R0 -vortex w0 := (A0 , u0 ) ∈ WC\Z , and passing to some 2,p (C \ Z, G), such that the subsequence, there exist gauge transformations gν ∈ Wloc following conditions are satisﬁed. (i) If R0 < ∞ then gν∗ wν converges to w0 in C ∞ on every compact subset of C \ Z.

46). We prove statement (ii). Assume that there exists a compact subset Q ⊆ Ω ν such that supν ||eR wν ||C 0 (Q) = ∞. Let zν ∈ Q be such that fν (zν ) → ∞. We choose a pair (r0 , w0 ) as in the claim. 47) and Remark 43 24 Here 25 see the norm is taken with respect to the metric ω(·, J·) on M . g. 2] 42 FABIAN ZILTENER (in the case r0 = ∞), we have E r0 (w0 ) ≥ Emin . 44) follows. This proves (ii) and concludes the proof of Proposition 40. We are now ready to prove Proposition 37 (p. 33). ν Proof of Proposition 37.

62) supz,z E w, A(ar, a−1 R) ≤ 4a−2+ε E(w), ∀a ≥ 2, −1+ε ¯ E(w), ∀a ≥ 4. 1]. 2). 2). 62) also uses the following remark. 46. Remark. Let M, ·, · M be a Riemannian manifold, G a compact Lie group that acts on M by isometries, P a G-bundle over [0, 1] 28 , A ∈ A(P ) a ∞ (P, M ) a map. We deﬁne connection, and u ∈ CG 1 |dA u|dt, (A, u) := 0 where dA u = du + Lu A, and the norm is taken with respect to the standard metric on [0, 1] and ·, · M . Furthermore, we deﬁne u ¯ : [0, 1] → M/G, u ¯(t) := Gu(p), where p ∈ P is any point over t.