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 .

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.

Totally Real Minimal Surfaces with Non-Circular Ellipse of Curvature in the Nearly Kähler S 6

1997 ◽  
Vol 56 (3) ◽  
pp. 625-644 ◽  
Author(s):  
John Bolton ◽  
Luc Vrancken ◽  
Lyndon M. Woodward
Keyword(s):  
Minimal Surfaces ◽  
Nearly Kähler ◽  
Ellipse Of Curvature ◽  
Totally Real

scholarly journals Trigonal minimal surfaces in flat tori

2007 ◽  
Vol 232 (2) ◽  
pp. 401-422
Author(s):  
Toshihiro Shoda
Keyword(s):  
Minimal Surfaces ◽  
Flat Tori

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 A Lehto–Virtanen-type theorem and a rescaling principle for an isolated essential singularity of a holomorphic curve in a complex space

2015 ◽  
Vol 26 (06) ◽  
pp. 1541009
Author(s):  
Yûsuke Okuyama
Keyword(s):  
Complex Space ◽  
Type Theorem ◽  
Essential Singularity ◽  
Holomorphic Curves ◽  
Holomorphic Curve ◽  
Picard Type Theorem

We establish a Lehto–Virtanen-type theorem and a rescaling principle for an isolated essential singularity of a holomorphic curve in a complex space, which are useful for establishing a big Picard-type theorem and a big Brody-type one for holomorphic curves.

Sign in / Sign up

Export Citation Format

Share Document