scholarly journals The Adjunction Inequality for Weyl-Harmonic Maps

2020 ◽  
Vol 7 (1) ◽  
pp. 129-140
Author(s):  
Robert Ream
Keyword(s):  
Minimal Surfaces ◽  
Twistor Space ◽  
Harmonic Maps ◽  
Holomorphic Curves ◽  
Weyl Connection ◽  
Almost Kähler ◽  
Conformal Manifolds

AbstractIn this paper we study an analog of minimal surfaces called Weyl-minimal surfaces in conformal manifolds with a Weyl connection (M4, c, D). We show that there is an Eells-Salamon type correspondence between nonvertical 𝒥-holomorphic curves in the weightless twistor space and branched Weyl-minimal surfaces. When (M, c, J) is conformally almost-Hermitian, there is a canonical Weyl connection. We show that for the canonical Weyl connection, branched Weyl-minimal surfaces satisfy the adjunction inequality\chi \left( {{T_f}\sum } \right) + \chi \left( {{N_f}\sum } \right) \le \pm {c_1}\left( {f*{T^{\left( {1,0} \right)}}M} \right).The ±J-holomorphic curves are automatically Weyl-minimal and satisfy the corresponding equality. These results generalize results of Eells-Salamon and Webster for minimal surfaces in Kähler 4-manifolds as well as their extension to almost-Kähler 4-manifolds by Chen-Tian, Ville, and Ma.

scholarly journals Transverse J-holomorphic curves in nearly Kähler $$\mathbb {CP}^3$$

2021 ◽  
Author(s):  
Benjamin Aslan
Keyword(s):  
Minimal Surfaces ◽  
Type Theorem ◽  
Holomorphic Curves ◽  
Nearly Kähler ◽  
Flat Tori

AbstractJ-holomorphic curves in nearly Kähler $$\mathbb {CP}^3$$ CP 3 are related to minimal surfaces in $$S^4$$ S 4 as well as associative submanifolds in $$\Lambda ^2_-(S^4)$$ Λ - 2 ( S 4 ) . We introduce the class of transverse J-holomorphic curves and establish a Bonnet-type theorem for them. We classify flat tori in $$S^4$$ S 4 and construct moment-type maps from $$\mathbb {CP}^3$$ CP 3 to relate them to the theory of $$\mathrm {U}(1)$$ U ( 1 ) -invariant minimal surfaces on $$S^4$$ S 4 .

Equi-harmonic maps and Plücker formulae for horizontal-holomorphic curves on flag manifolds

2013 ◽  
Vol 193 (4) ◽  
pp. 1089-1102
Author(s):  
Lino Grama ◽  
Caio J. C. Negreiros ◽  
Luiz A. B. San Martin
Keyword(s):  
Harmonic Maps ◽  
Holomorphic Curves ◽  
Flag Manifolds

scholarly journals Interpolation by conformal minimal surfaces and directed holomorphic curves

2019 ◽  
Vol 12 (2) ◽  
pp. 561-604 ◽  
Author(s):  
Antonio Alarcón ◽  
Ildefonso Castro-Infantes
Keyword(s):  
Minimal Surfaces ◽  
Holomorphic Curves

scholarly journals Energy of sections of the Deligne–Hitchin twistor space

2020 ◽  
Author(s):  
Florian Beck ◽  
Sebastian Heller ◽  
Markus Röser
Keyword(s):  
Moduli Space ◽  
Compact Riemann Surface ◽  
Twistor Space ◽  
Harmonic Maps ◽  
Energy Functional ◽  
Conformal Map ◽  
Twistor Spaces ◽  
Holomorphic Sections ◽  
Meromorphic Connection ◽  
Willmore Energy

Abstract We study a natural functional on the space of holomorphic sections of the Deligne–Hitchin moduli space of a compact Riemann surface, generalizing the energy of equivariant harmonic maps corresponding to twistor lines. We show that the energy is the residue of the pull-back along the section of a natural meromorphic connection on the hyperholomorphic line bundle recently constructed by Hitchin. As a byproduct, we show the existence of a hyper-Kähler potentials for new components of real holomorphic sections of twistor spaces of hyper-Kähler manifolds with rotating $$S^1$$ S 1 -action. Additionally, we prove that for a certain class of real holomorphic sections of the Deligne–Hitchin moduli space, the energy functional is basically given by the Willmore energy of corresponding equivariant conformal map to the 3-sphere. As an application we use the functional to distinguish new components of real holomorphic sections of the Deligne–Hitchin moduli space from the space of twistor lines.

TWISTORS, YANG-MILLS AMPLITUDES AND LANDAU LEVELS

2005 ◽  
Vol 20 (27) ◽  
pp. 6107-6121
Author(s):  
V. P. NAIR
Keyword(s):  
Twistor Space ◽  
Point Of View ◽  
Wave Functions ◽  
Landau Levels ◽  
Holomorphic Curves ◽  
Short Discussion ◽  
Yang Mills ◽  
Lowest Landau Level ◽  
Recent Developments ◽  
Discussion Review

We give a short discussion/review of the recent developments expressing the perturbative scattering amplitudes in Yang-Mills theory, specifically for the [Formula: see text] theory, in terms of holomorphic curves in a supersymmetric twistor space. Holomorphic curves, which are maps of CP1 to the supertwistor space, can also be interpreted as the lowest Landau level wave functions; this point of view is also briefly explained.

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