Harmonic mappings between surfaces : some local and global properties
Wood, John C. (1974) Harmonic mappings between surfaces : some local and global properties. PhD thesis, University of Warwick.
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Official URL: http://webcat.warwick.ac.uk/record=b1747082~S1
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|
|Institution:||University of Warwick|
|Theses Department:||Mathematics Institute|
|Supervisor(s)/Advisor:||Eells, James, 1926-2007 ; Lusztig, George, 1946-|
|Sponsors:||Science Research Council (Great Britain) (SRC)|
|Extent:||xiii, 166 p.|
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