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Isotropic harmonic maps to Kähler manifolds and related properties

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Glazebrook, James F. (1984) Isotropic harmonic maps to Kähler manifolds and related properties. PhD thesis, University of Warwick.

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Official URL: http://webcat.warwick.ac.uk/record=b3255056~S15

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Abstract

Taking complex projective space of n dimensions, CPn, with its Fubini-Study metric, Eells and Wood in [35] describe in detail, a bijective correspondence between full, holomorphic maps f:M CPn where M is a Riemann surface (open or closed) and isotropic harmonic maps ¢:M - CPn (see below). Their main result, broadly stated, is as follows:

Let L -CPn be the universal line bundle; we can define a universal lift ¢ of a smooth map ¢:M - CPn, as a section of the bundle Hom(¢-1,Cn+1)-M (here Cn+1 denotes the trivial (n+l)-plane bundle). If we take D to denote covariant differentiation in this bundle, then D splits into complex types D' and D". The harmonicity equation for ¢ is

D"D'¢+|D'¢|2¢=0.

With respect to the Hermitian inner product <,>, we say that ¢ is isotropic if

<D,a¢,D"B¢#> = 0 for all ¢,B with ¢+B>1 •

We say that a map into CPn is full, if its image lies in no proper projective subspace.

Item Type: Thesis (PhD)
Subjects: Q Science > QA Mathematics
Library of Congress Subject Headings (LCSH): Harmonic maps, Kählerian manifolds
Official Date: May 1984
Dates:
DateEvent
May 1984UNSPECIFIED
Institution: University of Warwick
Theses Department: Mathematics Institute
Thesis Type: PhD
Publication Status: Unpublished
Supervisor(s)/Advisor: Eells, James, 1926-2007
Sponsors: Science and Engineering Research Council (Great Britain)
Format of File: pdf
Extent: viii, 201 leaves : illustrations
Language: eng

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