Wood, John C. (1974) Harmonic mappings between surfaces : some local and global properties. PhD thesis, University of Warwick.

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Abstract

We develop some local and global properties of a harmonic map f :M -> N
between surfaces. Our first main result is a local description of the
possible singularities of such a harmonic map - we find there are four
types: degeneracy, general fold, meeting point of general folds, branch
point. As a corollary we have a result of Lewy and Heinz [Le1], [He1]. We
show that the singularities of a harnonic map in higher dimensions can be
qualitatively much nastier. We prove that there exist harmonic maps between
compact surfaces exhibiting general folds.
Our second main result is an inequality arising from the Gauss-Bonnet
formula relating the total curvature of the image of a harmonic map to its
Euler Characteristic. We derive some corollaries of this inequality and
compare with results obtained by convex function methods of Gordon [Go].
We also use the Gauss-Bonnet formula to show that if the codomain N has
negative curvature, certain types of unnecessary or redundant folding
cannot occur.
Other results include a characterisation of harmonic ramified coverings,
an upper bound on the number of zeros of the derivative of a harmonic map,
upper bounds on the number of singularities of different types for a
harmonic map into the flat torus, a study of holomorphic maps into the
sphere - such maps are harmonic [E-S], and some reflection principles for
harmonic maps.

Item Type:	Thesis or Dissertation (PhD)
Subjects:	Q Science > QA Mathematics
Library of Congress Subject Headings (LCSH):	Harmonic maps
Official Date:	June 1974
Dates:	Date Event June 1974 Submitted
Institution:	University of Warwick
Theses Department:	Mathematics Institute
Thesis Type:	PhD
Publication Status:	Unpublished
Supervisor(s)/Advisor:	Eells, James, 1926-2007 ; Lusztig, George, 1946-
Sponsors:	Science Research Council (Great Britain) (SRC)
Extent:	xiii, 166 p.
Language:	eng

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