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

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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:	Date Event May 1984 UNSPECIFIED
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Publication Status:	Unpublished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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