Canonical Metrics in Kähler Geometry:
Gespeichert in:
| 1. Verfasser: | |
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| Format: | Elektronisch E-Book |
| Sprache: | English |
| Veröffentlicht: |
Basel
Birkhäuser Basel
2000
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| Schriftenreihe: | Lectures in Mathematics. ETH Zürich, Department of Mathematics Research Institute of Mathematics
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| Schlagworte: | |
| Online-Zugang: | Volltext |
| Beschreibung: | This monograph results from the author's lectures at the ETH during the Spring Semester of 1997, when he was presenting a Nachdiplom course on KahlerEinstein metrics in complex differential geometry. There has been fundamental progress in complex differential geometry in the last two decades. The uniformization theory of canonical Kahler metrics has been established in higher dimensions. Many applications have been found. One manifestation of this is the use of Calabi-Yau spaces in the superstring theory. The aim of this monograph is to give an essentially self-contained introduction to the theory of canonical Kahler metrics on complex manifolds. It is also the author's hope to present the readers with some advanced topics in complex differential geometry which are hard to be found elsewhere. The topics include Calabi-Futaki invariants, Extremal Kahler metrics, the Calabi-Yau theorem on existence of Kahler Ricci-flat metrics, and recent progress on Kahler-Einstein metrics with positive scalar curvature. Applications of Kahler-Einstein metrics to the uniformization theory are also discussed. Readers with a good general knowledge in differential geometry and partial differential equations should be able to understand the materials in this monograph. I would like to thank the ETH for the opportunity to deliver the lectures in a very stimulating environment. In particular, I thank Meike Akveld for her patience and efficiency in taking notes of the lectures and producing the beautiful Jb.1EX file. Without her efforts, this monograph could never have been as it is now. I would also like to thank Ms. Nini Wong for her endless patience in proof-reading and correcting numerous typos in earlier versions of this monograph. Part of my work involved in this monograph was supported by National Science Foundation Grants DMS-9303999 and DMS-9802479, at Courant Institute of Mathematical Sciences and Massachusetts Institute of Technology. My research was also supported by a Simons Chair Fund at Massachusetts Institute of Technology. MIT, April 1999. Gang Tian Chapter1 Introduction to Kahler manifolds 1.1 Kahler metrics Let M be a compact Coo manifold. A Riemannian metricgon M is a smooth section of T*M @T*M defining a positive definite symmetric bilinear form on TxM for each x E M. In local coordinates Xl,...,X , one has a natural local n basis -iL,...,jL forTM, then g is represented by a smooth matrix-valued UXI UX n function {gij},where gij=g(a~i'a~j) . Note that {gij} is positive definite. The pair (M,g) is usually called a Riemannian manifold. Recall that an almost complex structureJ onM is a bundle automorphism of the tangent bundle TM satisfying j2 = - id. Definition1.1 The Nijenhuis tensor N(J) :TM xTM-+TM is given by N(v,w) = [v,w]+J[Jv,w]+J[v,Jw]- [Jv,Jw] for v,w vectorfields on M. An almost complex structure J on M is called integrable if there is a holomorphic structure (that is a set of charts with holomorphic transitionfunctions) such that J corresponds to the induced complex multiplication in TM x C. Clearly, any complex structure induces an integrable almost complex structure. The following theorem is due to Newlander and Nirenberg, see for example Appendix 8 in [14] |
| Beschreibung: | 1 Online-Ressource (VII, 101p) |
| ISBN: | 9783034883894 9783764361945 |
| DOI: | 10.1007/978-3-0348-8389-4 |
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| 500 | |a This monograph results from the author's lectures at the ETH during the Spring Semester of 1997, when he was presenting a Nachdiplom course on KahlerEinstein metrics in complex differential geometry. There has been fundamental progress in complex differential geometry in the last two decades. The uniformization theory of canonical Kahler metrics has been established in higher dimensions. Many applications have been found. One manifestation of this is the use of Calabi-Yau spaces in the superstring theory. The aim of this monograph is to give an essentially self-contained introduction to the theory of canonical Kahler metrics on complex manifolds. It is also the author's hope to present the readers with some advanced topics in complex differential geometry which are hard to be found elsewhere. The topics include Calabi-Futaki invariants, Extremal Kahler metrics, the Calabi-Yau theorem on existence of Kahler Ricci-flat metrics, and recent progress on Kahler-Einstein metrics with positive scalar curvature. Applications of Kahler-Einstein metrics to the uniformization theory are also discussed. | ||
| 500 | |a Readers with a good general knowledge in differential geometry and partial differential equations should be able to understand the materials in this monograph. I would like to thank the ETH for the opportunity to deliver the lectures in a very stimulating environment. In particular, I thank Meike Akveld for her patience and efficiency in taking notes of the lectures and producing the beautiful Jb.1EX file. Without her efforts, this monograph could never have been as it is now. I would also like to thank Ms. Nini Wong for her endless patience in proof-reading and correcting numerous typos in earlier versions of this monograph. Part of my work involved in this monograph was supported by National Science Foundation Grants DMS-9303999 and DMS-9802479, at Courant Institute of Mathematical Sciences and Massachusetts Institute of Technology. My research was also supported by a Simons Chair Fund at Massachusetts Institute of Technology. MIT, April 1999. | ||
| 500 | |a Gang Tian Chapter1 Introduction to Kahler manifolds 1.1 Kahler metrics Let M be a compact Coo manifold. A Riemannian metricgon M is a smooth section of T*M @T*M defining a positive definite symmetric bilinear form on TxM for each x E M. In local coordinates Xl,...,X , one has a natural local n basis -iL,...,jL forTM, then g is represented by a smooth matrix-valued UXI UX n function {gij},where gij=g(a~i'a~j) . Note that {gij} is positive definite. The pair (M,g) is usually called a Riemannian manifold. Recall that an almost complex structureJ onM is a bundle automorphism of the tangent bundle TM satisfying j2 = - id. Definition1.1 The Nijenhuis tensor N(J) :TM xTM-+TM is given by N(v,w) = [v,w]+J[Jv,w]+J[v,Jw]- [Jv,Jw] for v,w vectorfields on M. An almost complex structure J on M is called integrable if there is a holomorphic structure (that is a set of charts with holomorphic transitionfunctions) such that J corresponds to the induced complex multiplication in TM x C. Clearly, any complex structure induces an integrable almost complex structure. | ||
| 500 | |a The following theorem is due to Newlander and Nirenberg, see for example Appendix 8 in [14] | ||
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Datensatz im Suchindex
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| any_adam_object | |
| author | Tian, Gang |
| author_facet | Tian, Gang |
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| author_sort | Tian, Gang |
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| discipline | Mathematik |
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| spelling | Tian, Gang Verfasser aut Canonical Metrics in Kähler Geometry by Gang Tian Basel Birkhäuser Basel 2000 1 Online-Ressource (VII, 101p) txt rdacontent c rdamedia cr rdacarrier Lectures in Mathematics. ETH Zürich, Department of Mathematics Research Institute of Mathematics This monograph results from the author's lectures at the ETH during the Spring Semester of 1997, when he was presenting a Nachdiplom course on KahlerEinstein metrics in complex differential geometry. There has been fundamental progress in complex differential geometry in the last two decades. The uniformization theory of canonical Kahler metrics has been established in higher dimensions. Many applications have been found. One manifestation of this is the use of Calabi-Yau spaces in the superstring theory. The aim of this monograph is to give an essentially self-contained introduction to the theory of canonical Kahler metrics on complex manifolds. It is also the author's hope to present the readers with some advanced topics in complex differential geometry which are hard to be found elsewhere. The topics include Calabi-Futaki invariants, Extremal Kahler metrics, the Calabi-Yau theorem on existence of Kahler Ricci-flat metrics, and recent progress on Kahler-Einstein metrics with positive scalar curvature. Applications of Kahler-Einstein metrics to the uniformization theory are also discussed. Readers with a good general knowledge in differential geometry and partial differential equations should be able to understand the materials in this monograph. I would like to thank the ETH for the opportunity to deliver the lectures in a very stimulating environment. In particular, I thank Meike Akveld for her patience and efficiency in taking notes of the lectures and producing the beautiful Jb.1EX file. Without her efforts, this monograph could never have been as it is now. I would also like to thank Ms. Nini Wong for her endless patience in proof-reading and correcting numerous typos in earlier versions of this monograph. Part of my work involved in this monograph was supported by National Science Foundation Grants DMS-9303999 and DMS-9802479, at Courant Institute of Mathematical Sciences and Massachusetts Institute of Technology. My research was also supported by a Simons Chair Fund at Massachusetts Institute of Technology. MIT, April 1999. Gang Tian Chapter1 Introduction to Kahler manifolds 1.1 Kahler metrics Let M be a compact Coo manifold. A Riemannian metricgon M is a smooth section of T*M @T*M defining a positive definite symmetric bilinear form on TxM for each x E M. In local coordinates Xl,...,X , one has a natural local n basis -iL,...,jL forTM, then g is represented by a smooth matrix-valued UXI UX n function {gij},where gij=g(a~i'a~j) . Note that {gij} is positive definite. The pair (M,g) is usually called a Riemannian manifold. Recall that an almost complex structureJ onM is a bundle automorphism of the tangent bundle TM satisfying j2 = - id. Definition1.1 The Nijenhuis tensor N(J) :TM xTM-+TM is given by N(v,w) = [v,w]+J[Jv,w]+J[v,Jw]- [Jv,Jw] for v,w vectorfields on M. An almost complex structure J on M is called integrable if there is a holomorphic structure (that is a set of charts with holomorphic transitionfunctions) such that J corresponds to the induced complex multiplication in TM x C. Clearly, any complex structure induces an integrable almost complex structure. The following theorem is due to Newlander and Nirenberg, see for example Appendix 8 in [14] Mathematics Global analysis Global differential geometry Mathematical physics Differential Geometry Global Analysis and Analysis on Manifolds Mathematical Methods in Physics Mathematik Mathematische Physik Kähler-Mannigfaltigkeit (DE-588)4162978-4 gnd rswk-swf Differentialgeometrie (DE-588)4012248-7 gnd rswk-swf Komplexe Differentialgeometrie (DE-588)4193770-3 gnd rswk-swf Kähler-Einstein-Metrik (DE-588)4193456-8 gnd rswk-swf Komplexe Differentialgeometrie (DE-588)4193770-3 s Kähler-Mannigfaltigkeit (DE-588)4162978-4 s Kähler-Einstein-Metrik (DE-588)4193456-8 s 1\p DE-604 Differentialgeometrie (DE-588)4012248-7 s 2\p DE-604 https://doi.org/10.1007/978-3-0348-8389-4 Verlag Volltext 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Tian, Gang Canonical Metrics in Kähler Geometry Mathematics Global analysis Global differential geometry Mathematical physics Differential Geometry Global Analysis and Analysis on Manifolds Mathematical Methods in Physics Mathematik Mathematische Physik Kähler-Mannigfaltigkeit (DE-588)4162978-4 gnd Differentialgeometrie (DE-588)4012248-7 gnd Komplexe Differentialgeometrie (DE-588)4193770-3 gnd Kähler-Einstein-Metrik (DE-588)4193456-8 gnd |
| subject_GND | (DE-588)4162978-4 (DE-588)4012248-7 (DE-588)4193770-3 (DE-588)4193456-8 |
| title | Canonical Metrics in Kähler Geometry |
| title_auth | Canonical Metrics in Kähler Geometry |
| title_exact_search | Canonical Metrics in Kähler Geometry |
| title_full | Canonical Metrics in Kähler Geometry by Gang Tian |
| title_fullStr | Canonical Metrics in Kähler Geometry by Gang Tian |
| title_full_unstemmed | Canonical Metrics in Kähler Geometry by Gang Tian |
| title_short | Canonical Metrics in Kähler Geometry |
| title_sort | canonical metrics in kahler geometry |
| topic | Mathematics Global analysis Global differential geometry Mathematical physics Differential Geometry Global Analysis and Analysis on Manifolds Mathematical Methods in Physics Mathematik Mathematische Physik Kähler-Mannigfaltigkeit (DE-588)4162978-4 gnd Differentialgeometrie (DE-588)4012248-7 gnd Komplexe Differentialgeometrie (DE-588)4193770-3 gnd Kähler-Einstein-Metrik (DE-588)4193456-8 gnd |
| topic_facet | Mathematics Global analysis Global differential geometry Mathematical physics Differential Geometry Global Analysis and Analysis on Manifolds Mathematical Methods in Physics Mathematik Mathematische Physik Kähler-Mannigfaltigkeit Differentialgeometrie Komplexe Differentialgeometrie Kähler-Einstein-Metrik |
| url | https://doi.org/10.1007/978-3-0348-8389-4 |
| work_keys_str_mv | AT tiangang canonicalmetricsinkahlergeometry |