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By the conjecture of Calabi [45] proved by Yau [293, 295], there exists on every Calabi-Yau manifold a K ̈ahler metric with vanishing Ricci curvature. Currently, research on Calabi-Yau manifolds is a central focus in both mathematics and mathematical physics.
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16 feb 2006 · View a PDF of the paper titled Perspectives on geometric analysis, by Shing-Tung Yau View PDF Abstract: This is a survey paper on several aspects of differential geometry for the last 30 years, especially in those areas related to non-linear analysis.
ISBN: 978-1-57146-377-7; 978-1-57146-368-5. [565] Schoen, Richard; Yau, Shing Tung: Harmonic maps and the topology of stable hypersurfaces and manifolds with non-negative Ricci curvature. International Press, Boston, MA, 2019, 187–195.
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- 1 Introduction
- Profound open problems raised.
- 4 Basics on Kahler-Einstein metrics and Calabi conjectures
- 5 Some applications of Kahler-Einstein metrics and Calabi-Yau manifolds
- 6 Harmonic maps
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- 14 Partial list of papers and books of Yau
- 15 Papers and books by others
Abstract The purpose of this article is to give a brief overview of the work of Yau by compiling lists of his papers and books together with brief comments on some part of his work, and describing a sample of some open problems raised by him. It tries to complement other more detailed commentaries on particular aspects of his work by experts in thi...
Shing-Tung Yau has revolutionized the broad eld of geometric analysis by combining partial di er-ential equations with di erential geometry and applying geometric analysis to algebraic geometry. Besides his celebrated solution to the Calabi conjecture on Kahler-Einstein metrics and applications of the Calabi-Yau manifolds to the string theory and t...
Yau has raised many important questions and made many conjectures. Several of them have lead to a lot of development and opened up new elds: Uniformization of complex manifolds. Sizes on the space of harmonic functions with polynomial growth. Rank rigidity and geometry of nonpositively curved compact or nite volume manifolds. Relations between Kahl...
It is known that any smooth manifold M admits a Riemannian metric. A natural and basic question is whether there is any best or canonical Riemannian metric on it. When M is an orientable surface, by the uniformization theorem, M always admits a Rieman-nian metric of constant curvature. If M is given a complex structure, then there is also a metric ...
If a complex manifold admits a canonical Kahler-Einstein metric, the metric can be used to un-derstand the geometry and topology of the manifold in a very e ective way. The results obtained so far seem to indicate that this has mainly been the case when the Ricci curvature (or the rst Chern class) is nonpositive. For the convenience of the reader, ...
Harmonic maps are one of the analytical tools that have been systematically developed and applied by Yau to a wide range of questions in algebraic, Kahler and Riemannian geometry. Let us recall some basic notations: Let f : M ! N be a (smooth) map between Riemannian manifolds M and N. Let e(f) = trace of the pull-back metric f dsN 2 relative to the...
where [M] denotes the fundamental homology class of the domain M, [M] its Kahler class and c1(M) its rst Chern class. The image N is always assumed to be a compact quotient of polydiscs with the usual induced metric and complex structure; moreover, the functional determinant of f : M ! N is not identically zero. Under the same condition, the paper ...
M A critical point of this functional is called an extremal metric by Calabi, and any Kahler-Einstein metric is an extremal metric. There has been a lot of work related to this conjecture, for example, this conjecture has been made more precise and extended to study Kahler metrics of constant scalar curvature and many di erent notions of stability ...
To the best of our knowledge, we have put down most of the published papers and books of Yau according to MathSciNet. Since he has written so many and some papers are not even listed in MathSciNet, some might have slipped through.
This part of the references contain papers and books cited or related to results discussed in this article.
The Shape of a Life. one mathematician’s search . for the universe’s . hidden geometry. new haven and london. Published with assistance from the foundation established in memory of William McKean Brown. Copyright © 2019 by Shing-Tung Yau and Steve Nadis. All rights reserved.
Republication, systematic copying, or mass reproduction of any material in this publication is permitted only under license from International Press. ISBN 978-1-57146-198-8. Paperback reissue 2010. This is a facsimile reproduction of the original work as it was published in 1994 under ISBN 1-57146-012-8 (clothbound).
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In. their new book, The Shape of Inner Space, Shing-Tung Yau and Steve Nadis review some beautiful developments in geometry and mathematical physics involving a special class of spaces known...
