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Geometry of Ricci-flat Kähler manifolds and some counterexamples

Author(s)
Božin, Vladimir, 1973-
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Massachusetts Institute of Technology. Dept. of Mathematics.
Advisor
Gang Tian.
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M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission. http://dspace.mit.edu/handle/1721.1/7582
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Abstract
In this work, we study geometry of Ricci-flat Kähler manifolds, and also provide some counterexample constructions. We study asymptotic behavior of complete Ricci-flat metrics at infinity and consider a construction of approximate Ricci-flat metrics on quasiprojective manifolds with a divisor with normal crossings removed, by means of reducing torsion of a non-Kähler metric with the right volume form. Next, we study special Lagrangian fibrations using methods of geometric function theory. In particular, we generalize the method of extremal length and prove a generaliziation of the Teichmiiller theorem. We relate extremal problems to the existence of special Lagrangian fibrations in the large complex structure limit of Calabi-Yau manifolds. We proceed to some problems in the theory of minimal surfaces, disproving the Schoen-Yau conjecture and providing a first example of a proper harmonic map from the unit disk to a complex plane. In the end, we prove that the union closed set conjecture is equivalent to a strengthened version, giving a construction which might lead to a counterexample.
Description
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2004.
 
Includes bibliographical references (leaves 61-64).
 
Date issued
2004
URI
http://hdl.handle.net/1721.1/32243
Department
Massachusetts Institute of Technology. Department of Mathematics
Publisher
Massachusetts Institute of Technology
Keywords
Mathematics.

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