Twistorial Construction of Harmonic Maps of Surfaces into Four-Manifolds

Eells, James; Salamon, Simon

doi:10.1142/9789814360197_0021

Cited by 121 publications

(215 citation statements)

References 4 publications

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“…Moreover, the Levi-Civita connection ∇ of (M, g) induces a splitting T Z = H ⊕ V of the tangent bundle of Z into horizontal and vertical components, so that H ∼ = p −1 * T M acquires a tautological complex structure J H given by J H x,j = j for x ∈ M and j ∈ p −1 (x) ∼ = CP 1 . Following [35], we define an almost-complex structure J on Z, by…”

Section: Twistorial Examplesmentioning

confidence: 99%

The curvature and the integrability of almost-Kähler manifolds: A survey

Apostolov¹,

Drăghici²

2003

Symplectic and Contact Topology: Interactions and Perspectives

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Section: Twistorial Examplesmentioning

confidence: 99%

The curvature and the integrability of almost-Kähler manifolds: A survey

Apostolov¹,

Drăghici²

2003

Symplectic and Contact Topology: Interactions and Perspectives

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“…For n ∈ N , let J n be the compatible complex structure on T φ(n) M that restricts to J on T φ(n) N . We define a twistor lift [8] of N to Z asφ + : N → Z; n → J n . This map can be broken down into two components: a map sending n to the 2-plane T φ(n) N ⊂ T φ(n) M , followed by a map sending T φ(n) N to J n .…”

Section: Tensors On Orbifoldsmentioning

confidence: 99%

“…There are two types of coordinate transformation that are permissible: firstly the map z → z −1 , which interchanges 0 and ∞; and secondly, the map z → λ.z for λ ∈ C * , which fixes 0 and ∞. When L ∈ M • , we require |λ| = 1 to preserve equation (8). Figure 7 depicts a real principal line (when k = 3).…”

Section: The Intersection Numbermentioning

confidence: 99%

Compact anti-self-dual orbifolds with torus actions

Wright

2010

Sel. Math. New Ser.

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Abstract. We give a classification of toric anti-self-dual conformal structures on compact 4-orbifolds with positive Euler characteristic. Our proof is twistor theoretic: the interaction between the complex torus orbits in the twistor space and the twistor lines induces meromorphic data, which we use to recover the conformal structure. A compact anti-self-dual orbifold can also be constructed by adding a point at infinity to an asymptotically locally Euclidean (ALE) scalar-flat Kähler orbifold. We use this observation to classify ALE scalar-flat Kähler 4-orbifolds whose isometry group contain a 2-torus.

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“…Then Tw(M) has a natural Kähler-Einstein structure (I + , g), obtained by interpreting unit vectors in 2 − (M) as complex structure operators on T M. Changing the sign of I + on T M, we obtain an almost complex structure I − which is also compatible with the metric g [Eells and Salamon 1985]. A straightforward computation insures that (Tw(M), I − , g) is nearly Kähler [Muškarov 1987].…”

Section: C Examples Of Nearly Kähler Manifoldsmentioning

confidence: 99%

An intrinsic volume functional on almost complex 6-manifolds and nearly Kähler geometry

Verbitsky¹

2008

Pacific J. Math.

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Let (M, I) be an almost complex 6-manifold. The obstruction to integrability of the almost complex structure (the so-called Nijenhuis tensor) N : Λ 0,1 (M ) −→ Λ 2,0 (M ) maps a 3-dimensional bundle to a 3-dimensional one. We say that Nijenhuis tensor is non-degenerate if it is an isomorphism. An almost complex manifold (M, I) is called nearly Kähler if it admits a Hermitian form ω such that ∇(ω) is totally antisymmetric, ∇ being the Levi-Civita connection. We show that a nearly Kähler metric on a given almost complex 6-manifold with non-degenerate Nijenhuis tensor is unique (up to a constant). We interpret the nearly Kähler property in terms of G 2 -geometry and in terms of connections with totally antisymmetric torsion, obtaining a number of equivalent definitions.We construct a natural diffeomorphism-invariant functional I −→ M Vol I on the space of almost complex structures on M , similar to the Hitchin functional, and compute its extrema in the following important case. Consider an almost complex structure I with non-degenerate Nijenhuis tensor, admitting a Hermitian connection with totally antisymmetric torsion. We show that the Hitchin-like functional I −→ M Vol I has an extremum in I if and only if (M, I) is nearly Kähler.

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Twistorial Construction of Harmonic Maps of Surfaces into Four-Manifolds

Cited by 121 publications

References 4 publications

The curvature and the integrability of almost-Kähler manifolds: A survey

The curvature and the integrability of almost-Kähler manifolds: A survey

Compact anti-self-dual orbifolds with torus actions

An intrinsic volume functional on almost complex 6-manifolds and nearly Kähler geometry

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