Harmonic maps of finite type

1997 ◽  
pp. 162-170
Keyword(s):  
Finite Type ◽  
Harmonic Maps

scholarly journals Properly embedded minimal annuli in $\mathbb{S}^2 \times \mathbb{R}$

2020 ◽  
Vol 5 (1) ◽  
Author(s):  
L Hauswirth ◽  
M Kilian ◽  
M U Schmidt
Keyword(s):  
Spectral Data ◽  
Finite Type ◽  
Harmonic Maps ◽  
Connected Component ◽  
Curvature Estimate ◽  
The Third ◽  
Minimal Annulus

Abstract We prove that every properly embedded minimal annulus in $\mathbb{S}^2\times\mathbb{R}$ is foliated by circles. We show that such minimal annuli are given by periodic harmonic maps $\mathbb{C} \to \mathbb{S}^2$ of finite type. Such harmonic maps are parameterized by spectral data, and we show that continuous deformations of the spectral data preserve the embeddedness of the corresponding annuli. A curvature estimate of Meeks and Rosenberg is used to show that each connected component of spectral data of embedded minimal annuli contains a maximum of the flux of the third coordinate. A classification of these maxima allows us to identify the spectral data of properly embedded minimal annuli with the spectral data of minimal annuli foliated by circles.

scholarly journals Finite type Lorentz harmonic maps and the method of Symes

2002 ◽  
Vol 17 (1) ◽  
pp. 43-53 ◽  
Author(s):  
Josef Dorfmeister ◽  
Ivan Sterling
Keyword(s):  
Finite Type ◽  
Harmonic Maps

scholarly journals ADDING A UNITON VIA THE DPW METHOD

2009 ◽  
Vol 20 (08) ◽  
pp. 997-1010 ◽  
Author(s):  
N. CORREIA ◽  
R. PACHECO
Keyword(s):  
Finite Type ◽  
Symmetric Spaces ◽  
Harmonic Maps ◽  
Harmonic Map ◽  
Point Of View ◽  
Type Property ◽  
Riemannian Symmetric Spaces

In this paper we describe how the operation of adding a uniton arises via the DPW method of obtaining harmonic maps into compact Riemannian symmetric spaces from certain holomorphic 1-forms. We exploit this point of view to investigate which unitons preserve finite type property of harmonic maps. In particular, we prove that the Gauss bundle of a harmonic map of finite type into a Grassmannian is also of finite type.

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.

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