2026-09-19T06:54:01Zhttps://dspace.mit.edu/server/oai/requestoai:dspace.mit.edu:1721.1/336002022-01-13T07:54:35Zcom_1721.1_7582com_1721.1_7581col_1721.1_131022
Shing-Tung Yau.
Wu, Damin, Ph. D. Massachusetts Institute of Technology
Massachusetts Institute of Technology. Dept. of Mathematics.
Massachusetts Institute of Technology. Department of Mathematics
2006-07-31T15:16:01Z
2006-07-31T15:16:01Z
2005
2005
http://hdl.handle.net/1721.1/33600
64035977
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2005.
Includes bibliographical references (p. 61-64).
In this thesis, we derive the asymptotic expansion of the Kiihler-Einstein metrics on certain quasi-projective varieties, which can be compactified by adding a divisor with simple normal crossings. The weighted Cheng-Yau Hilder spaces and the log-filtrations based on the bounded geometry are introduced to characterize the asymptotics. We first develop the analysis of the Monge-Ampere operators on these weighted spaces. We construct a family of linear elliptic operators which can be viewed as certain conjugacies of the specially linearized Monge-Ampbre operators. We derive a theorem of Fredholm alternative for such elliptic operators by the Schauder theory and Yau's generalized maximum principle. Together these results derive the isomorphism theorems of the Monge-Ampbre operators, which imply that the Monge-Ampere operators preserve the log-filtration of the Cheng-Yau Holder ring. Next, by choosing a canonical metric on the submanifold, we construct an initial Kidhler metric on the quasi-projective manifold such that the unique solution of the Monge-Ampere equation belongs to the weighted -1 Cheng-Yau Hölder ring. Moreover, we generalize the Fefferman's operator to act on the volume forms and obtain an iteration formula.
(cont.) Finally, with the aid of the isomorphism theorems and the iteration formula we derive the desired asymptotics from the initial metric. Furthermore, we prove that the obtained asymptotics is canonical in the sense that it is independent of the extensions of the canonical metric on the submanifold.
by Damin Wu.
Ph.D.
64 p.
2046820 bytes
2041635 bytes
application/pdf
application/pdf
eng
Massachusetts Institute of Technology
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http://dspace.mit.edu/handle/1721.1/7582
Mathematics.
Higher canonical asymptotics of Kähler-Einstein metrics on quasi-projective manifolds
Thesis
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<Abstract>In this thesis, we derive the asymptotic expansion of the Kiihler-Einstein metrics on certain quasi-projective varieties, which can be compactified by adding a divisor with simple normal crossings. The weighted Cheng-Yau Hilder spaces and the log-filtrations based on the bounded geometry are introduced to characterize the asymptotics. We first develop the analysis of the Monge-Ampere operators on these weighted spaces. We construct a family of linear elliptic operators which can be viewed as certain conjugacies of the specially linearized Monge-Ampbre operators. We derive a theorem of Fredholm alternative for such elliptic operators by the Schauder theory and Yau's generalized maximum principle. Together these results derive the isomorphism theorems of the Monge-Ampbre operators, which imply that the Monge-Ampere operators preserve the log-filtration of the Cheng-Yau Holder ring. Next, by choosing a canonical metric on the submanifold, we construct an initial Kidhler metric on the quasi-projective manifold such that the unique solution of the Monge-Ampere equation belongs to the weighted -1 Cheng-Yau Hölder ring. Moreover, we generalize the Fefferman's operator to act on the volume forms and obtain an iteration formula.</Abstract>
<Abstract>(cont.) Finally, with the aid of the isomorphism theorems and the iteration formula we derive the desired asymptotics from the initial metric. Furthermore, we prove that the obtained asymptotics is canonical in the sense that it is independent of the extensions of the canonical metric on the submanifold.</Abstract>
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