Geometry of Ricci-flat Kähler manifolds and some counterexamples
Name
56019201-MIT.pdf
Description
Full printable version
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2.51 MB
Format
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Checksum (MD5)
99eb9334b721c530ca6bb74561f87a58
Author(s)
Božin, Vladimir, 1973-
Advisor(s)
Gang Tian.
Date Issued
2004
Publisher
Massachusetts Institute of Technology
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).
Subjects
Mathematics.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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