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Liouville-Riemann-Roch Theorems on Abelian Coverings

  • Book
  • © 2021

Overview

Authors:
  1. Minh Kha

    1. Department of Mathematics, University of Arizona, Tucson, USA

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    You can also search for this author in PubMed   Google Scholar

  2. Peter Kuchment

    1. Department of Mathematics, Texas A&M University, College Station, USA

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    You can also search for this author in PubMed   Google Scholar

  • 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

Part of the book series: Lecture Notes in Mathematics (LNM, volume 2245)

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Table of contents (5 chapters)

  1. Front Matter

    Pages i-xii
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  2. Preliminaries

    • Minh Kha, Peter Kuchment
    Pages 1-21

  3. The Main Results

    • Minh Kha, Peter Kuchment
    Pages 23-33

  4. Proofs of the Main Results

    • Minh Kha, Peter Kuchment
    Pages 35-53

  5. Specific Examples of Liouville-Riemann-Roch Theorems

    • Minh Kha, Peter Kuchment
    Pages 55-66

  6. Auxiliary Statements and Proofs of Technical Lemmas

    • Minh Kha, Peter Kuchment
    Pages 67-84

  7. Back Matter

    Pages 85-96
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Keywords

About this book

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.

Authors and Affiliations

  • Department of Mathematics, University of Arizona, Tucson, USA

    Minh Kha

  • Department of Mathematics, Texas A&M University, College Station, USA

    Peter Kuchment

Bibliographic Information

  • Book Title: Liouville-Riemann-Roch Theorems on Abelian Coverings

  • Authors: Minh Kha, Peter Kuchment

  • Series Title: Lecture Notes in Mathematics

  • DOI: https://doi.org/10.1007/978-3-030-67428-1

  • Publisher: Springer Cham

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021

  • Softcover ISBN: 978-3-030-67427-4Published: 13 February 2021

  • eBook ISBN: 978-3-030-67428-1Published: 12 February 2021

  • Series ISSN: 0075-8434

  • Series E-ISSN: 1617-9692

  • Edition Number: 1

  • Number of Pages: XII, 96

  • Number of Illustrations: 1 b/w illustrations, 1 illustrations in colour

  • Topics: Global Analysis and Analysis on Manifolds, Analysis, Topology

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