Holomorphic maps into Grassmann manifolds (harmonic maps into Grassmann manifolds III)

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.

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Harmonic Maps and Holomorphic Maps

Geometry of Harmonic Maps ◽

10.1007/978-1-4612-4084-6_4 ◽

1996 ◽

pp. 97-119

Author(s):

Yuanlong Xin

Keyword(s):

Harmonic Maps ◽

Holomorphic Maps

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Regularity and Blow-Up Analysis for J-Holomorphic Maps

Communications in Contemporary Mathematics ◽

10.1142/s0219199703001063 ◽

2003 ◽

Vol 05 (04) ◽

pp. 671-704

Author(s):

Changyou Wang

Keyword(s):

Blow Up ◽

Harmonic Maps ◽

Hermitian Manifold ◽

Singular Set ◽

Holomorphic Maps ◽

Almost Hermitian Manifold ◽

Holomorphic Map ◽

Energy Quantization ◽

Blow Up Analysis ◽

Almost Kähler

If u∈H1(M,N) is a weakly J-holomorphic map from a compact without boundary almost hermitian manifold (M,j,g) into another compact without boundary almost hermitian manifold (N,J,h). Then it is smooth near any point x where Du has vanishing Morrey norm ℳ2,2m-2, with 2m= dim (M). Hence H2m-2measure of the singular set for a stationary J-holomorphic map is zero. Blow-up analysis and the energy quantization theorem are established for stationary J-holomorphic maps. Connections between stationary J-holomorphic maps and stationary harmonic maps are given for either almost Kähler manifolds M and N or symmetric ∇hJ.

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A SCHWARZ LEMMA FOR -HARMONIC MAPS AND THEIR APPLICATIONS

Bulletin of the Australian Mathematical Society ◽

10.1017/s000497271700051x ◽

2017 ◽

Vol 96 (3) ◽

pp. 504-512 ◽

Cited By ~ 3

Author(s):

QUN CHEN ◽

GUANGWEN ZHAO

Keyword(s):

Riemannian Manifolds ◽

Harmonic Maps ◽

Schwarz Lemma ◽

Kähler Manifolds ◽

Kahler Manifolds ◽

Holomorphic Maps ◽

Hermitian Manifolds ◽

Almost Hermitian Manifolds ◽

Almost Kähler

We establish a Schwarz lemma for $V$-harmonic maps of generalised dilatation between Riemannian manifolds. We apply the result to obtain corresponding results for Weyl harmonic maps of generalised dilatation from conformal Weyl manifolds to Riemannian manifolds and holomorphic maps from almost Hermitian manifolds to quasi-Kähler and almost Kähler manifolds.

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

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100000876 ◽

1983 ◽

Vol 94 (3) ◽

pp. 483-494 ◽

Cited By ~ 4

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].

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CRITICAL POINTS OF YANG–MILLS–HIGGS FUNCTIONAL

Communications in Contemporary Mathematics ◽

10.1142/s0219199711004403 ◽

2011 ◽

Vol 13 (03) ◽

pp. 463-486 ◽

Cited By ~ 4

Author(s):

CHONG SONG

Keyword(s):

Perturbation Method ◽

Critical Points ◽

Existence Result ◽

Harmonic Maps ◽

Fiber Bundles ◽

Energy Identity ◽

Holomorphic Maps ◽

Yang Mills ◽

The One

We use Sacks–Uhlenbeck's perturbation method to find critical points of the Yang–Mills–Higgs functional on fiber bundles with 2-dimensional base manifolds. Such critical points can be regarded as a generalization of harmonic maps from surfaces, and also a generalization of the so-called twisted holomorphic maps [15]. We prove an existence result analogous to the one for harmonic maps. In particular, we show that the so-called energy identity holds for the Yang–Mills–Higgs functional.

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