Liouville-Riemann-Roch Theorems on Abelian Coverings
Authors: Kha, Minh, Kuchment, Peter
Free Preview- The first unified exposition of Liouville and Riemann–Roch type theorems for elliptic operators on abelian coverings
- Gives a well-organized and self-contained exposition of the topic, including new results
- Intersects with geometric analysis, the spectral theory of periodic operators, and their applications
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- About this book
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This book is devoted to computing the index of elliptic PDEs on non-compact Riemannian manifolds in the presence of local singularities and zeros, as well as polynomial growth at infinity. The classical Riemann–Roch theorem and its generalizations to elliptic equations on bounded domains and compact manifolds, due to Maz’ya, Plameneskii, Nadirashvilli, Gromov and Shubin, account for the contribution to the index due to a divisor of zeros and singularities. On the other hand, the Liouville theorems of Avellaneda, Lin, Li, Moser, Struwe, Kuchment and Pinchover provide the index of periodic elliptic equations on abelian coverings of compact manifolds with polynomial growth at infinity, i.e. in the presence of a "divisor" at infinity.
A natural question is whether one can combine the Riemann–Roch and Liouville type results. This monograph shows that this can indeed be done, however the answers are more intricate than one might initially expect. Namely, the interaction between the finite divisor and the point at infinity is non-trivial.
The text is targeted towards researchers in PDEs, geometric analysis, and mathematical physics.
- Table of contents (5 chapters)
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Preliminaries
Pages 1-21
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The Main Results
Pages 23-33
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Proofs of the Main Results
Pages 35-53
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Specific Examples of Liouville-Riemann-Roch Theorems
Pages 55-66
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Auxiliary Statements and Proofs of Technical Lemmas
Pages 67-84
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Table of contents (5 chapters)
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Bibliographic Information
- Bibliographic Information
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- Book Title
- Liouville-Riemann-Roch Theorems on Abelian Coverings
- Authors
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- Minh Kha
- Peter Kuchment
- Series Title
- Lecture Notes in Mathematics
- Series Volume
- 2245
- Copyright
- 2021
- Publisher
- Springer International Publishing
- Copyright Holder
- The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG
- eBook ISBN
- 978-3-030-67428-1
- DOI
- 10.1007/978-3-030-67428-1
- Softcover ISBN
- 978-3-030-67427-4
- Series ISSN
- 0075-8434
- Edition Number
- 1
- Number of Pages
- XII, 96
- Number of Illustrations
- 1 b/w illustrations, 1 illustrations in colour
- Topics
