scholarly journals ON THE SPACE OF HARMONIC 2-SPHERES IN CP2

1996 ◽  
Vol 07 (02) ◽  
pp. 211-225 ◽  
Author(s):  
L. LEMAIRE ◽  
J.C. WOOD
Keyword(s):  
Harmonic Maps ◽  
Holomorphic Maps ◽  
Ramification Index ◽  
Gauss Transform ◽  
Complex Projective

Carrying further the work of T.A. Crawford, we show that each component of the space of harmonic maps from the 2-sphere to complex projective 2-space of degree d and energy 4πE is a smooth closed submanifold of the space of all Cj maps (j≥2). We achieve this by showing that the Gauss transform which relates them to spaces of holomorphic maps of given degree and ramification index is smooth and has injective differential.

Harmonic maps from surfaces into pseudo-Riemannian spheres and hyperbolic spaces

1983 ◽  
Vol 94 (3) ◽  
pp. 483-494 ◽  
Author(s):  
S. Erdem
Keyword(s):  
Projective Space ◽  
Complex Projective Space ◽  
Harmonic Maps ◽  
Hyperbolic Spaces ◽  
Euclidean Sphere ◽  
Holomorphic Maps ◽  
Space Forms ◽  
Indefinite Metrics ◽  
Complex Hyperbolic Spaces ◽  
Complex Projective

In [2, 4, 5, 6, 7] Calabi, Barbosa and Chern showedthat there is a 2:1 correspondence between arbitrary pairs of full isotropic (terminology as in [8]) harmonic maps ±φ:M→S2mfrom a Riemann surface to a Euclidean sphere and full totally isotropic holomorphic maps f:M→2mfrom the surface to complex projective space. In this paper we show, very explicitly, how to construct a similar one-to-one correspondence whenS2mis replaced by some other space forms of positive and negative curvatures with their standard (indefinite) metrics obtained by restricting a standard (indefinite) bilinear form on Euclidean space to the tangent spaces. We get over a difficulty encountered by Barbosa of dealing with the zeros of a certain wedge product by a technique adapted from [8]. The case of complex projective space forms (indefinite complex projective and complex hyperbolic spaces) will be considered in a separate paper. Some further developments in classification theorems are given by Eells and Wood [8], Rawnsley[14], [15] and Erdem and Wood [10].

The Space of Harmonic Maps from the 2-Sphere to the Complex Projective Plane

1997 ◽  
Vol 40 (3) ◽  
pp. 285-295 ◽  
Author(s):  
T. Arleigh Crawford
Keyword(s):  
Projective Plane ◽  
Harmonic Maps ◽  
Complex Manifolds ◽  
Complex Projective Plane ◽  
Fixed Degree ◽  
Complex Projective ◽  
Path Connected

AbstractIn this paper we study the topology of the space of harmonic maps from S2 to ℂℙ2.We prove that the subspaces consisting of maps of a fixed degree and energy are path connected. By a result of Guest and Ohnita it follows that the same is true for the space of harmonic maps to ℂℙn for n ≥ 2. We show that the components of maps to ℂℙ2 are complex manifolds.

scholarly journals Modified defect relations for the gauss map of minimal surfaces, III

1991 ◽  
Vol 124 ◽  
pp. 13-40 ◽  
Author(s):  
Hirotaka Fujimoto
Keyword(s):  
Minimal Surface ◽  
Complex Projective Space ◽  
Meromorphic Functions ◽  
Total Curvature ◽  
Gauss Map ◽  
Holomorphic Maps ◽  
Complete Minimal Surface ◽  
Defect Relation ◽  
Definition Of ◽  
Complex Projective

In [5], the author proved that the Gauss map of a nonflat complete minimal surface immersed in R3 can omit at most four points of the sphere, and in [7] he revealed some relations between this result and the defect relation in Nevanlinna theory on value distribution of meromorphic functions. Afterwards, Mo and Osserman obtained an improvement of these results in their paper [11], which asserts that if the Gauss map of a nonflat complete minimal surface M immersed in R3 takes on five distinct values only a finite number of times, then M has finite total curvature. The author also gave modified defect relations for holomorphic maps of a Riemann surface with a complete conformai metric into the n-dimensional complex projective space Pn(C) and, as its application, he showed that, if the (generalized) Gauss map G of a complete minimal surface M immersed in Rm is nondegenerate, namely, the image G(M) is not contained in any hyperplane in Pm − 1(C), then it can omit at most m(m + 1)/2 hyperplanes in general position ([8]). Here, the number m(m + 1)/2 is best-possible for arbitrary odd numbers and some small even numbers m (see [6]). Recently, Ru showed that the “nondegenerate” assumption of the above result can be dropped ([13]). In this paper, we shall introduce a new definition of modified defect and prove a refined Modified defect relation. As its application, we shall give some improvements of the above-mentioned results in [5], [7], [8], [11] and [13].

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