Pseudo-holomorphic maps and bubble trees

Parker, Thomas H.; Wolfson, Jon

doi:10.1007/bf02921330

Cited by 101 publications

(83 citation statements)

References 13 publications

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“…The first part follows directly from the Theorem 3.4. Compactness follows from the usual compactification theorem for holomorphic curves (see [9] and also [14], [30], [40]), which hold thanks to the energy bound (Lemma 3.5).…”

Section: Proposition 316 If J Is Sufficiently Stretched Out Along Tmentioning

confidence: 99%

Holomorphic disks and topological invariants for closed three-manifolds

2004

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Section: Proposition 316 If J Is Sufficiently Stretched Out Along Tmentioning

confidence: 99%

Holomorphic disks and topological invariants for closed three-manifolds

2004

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“…In fact, by following the procedure introduced in [12], one can modify the Sacks-Uhlenbeck renormalization and iterate, obtaining bubbles on bubbles. The set of all bubble maps then forms a "bubble tree" ( [12]). One would like to know in precisely what sense the sequence {f n } converges to this bubble tree.…”

Section: < Eomentioning

confidence: 99%

“…The bubble tree is constructed in §1 using the procedure of [12]. The construction is elementary and requires only the basic facts about harmonic maps stated in Proposition 1.1.…”

Section: < Eomentioning

confidence: 99%

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Bubble tree convergence for harmonic maps

Parker¹

1996

J. Differential Geom.

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155

164

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Let Σ be a compact Riemann surface. Any sequence f n : Σ -> M of harmonic maps with bounded energy has a "bubble tree limit" consisting of a harmonic map /o : Σ -> M and a tree of bubblesWe give a precise construction of this bubble tree and show that the limit preserves energy and homotopy class, and that the images of the f n converge pointwise. We then give explicit counterexamples showing that bubble tree convergence fails (i) for harmonic maps f n when the conformal structure of Σ varies with n, and (ii) when the conformal structure is fixed and {/ n } is a Palais-Smale sequence for the harmonic map energy.

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“…This theorem is due to Gromov [6]; proofs can be also found in [13] and [14]. We shall require a more precise description of Gromov convergency, which can be derived from [14].…”

Section: Fig 2 Annulusmentioning

confidence: 99%

Deformations of non-compact complex curves and envelopes of meromorphy of spheres

Ivashkovich¹,

Shevchishin²

1998

Sb. Math.

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Abstract. The paper is devoted to the properties of the envelopes of meromorphy of neighborhoods of symplectically immersed two-spheres in complex Kähler surfaces. The method used to study the envelopes of meromorphy is based on Gromov's theory of pseudoholomorphic curves. The exposition includes a construction of a complete family of holomorphic deformations of a non-compact complex curve in a complex manifold, parametrized by a finite codimension analytic subset of a Banach ball. The existence of this family is used to prove a generalization of Levi's continuity principle, which is applied to describe envelopes of meromorphy.Bibliography: 15 titles.

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Pseudo-holomorphic maps and bubble trees

Cited by 101 publications

References 13 publications

Holomorphic disks and topological invariants for closed three-manifolds

Holomorphic disks and topological invariants for closed three-manifolds

Bubble tree convergence for harmonic maps

Deformations of non-compact complex curves and envelopes of meromorphy of spheres

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