Geometry and Spectra of Compact Riemann Surfaces

Buser, Peter

ISBN 10: 0817634061 / ISBN 13: 9780817634063

Editore: Boston/Basel/Berlin : Birkhäuser, 1992

Riguardo questo articolo

Original boards, illustrated with numerous equations and diagrams, 8vo. Progress in Mathematics, 106; Name in pen on title page.

Codice articolo 342009-ZA22

Dati bibliografici

• Titolo: Geometry and Spectra of Compact Riemann Surfaces
• Editore: Boston/Basel/Berlin : Birkhäuser
• Data di pubblicazione: 1992
• Lingua: Inglese
• ISBN 10: 0817634061
• ISBN 13: 9780817634063
• Rilegatura: Rilegato
• Condizione: Good

Descrizione articolo

Riassunto

This monograph is a self-contained introduction to the geometry of Riemann Surfaces of constant curvature -1 and their length and eigenvalue spectra. It focuses on two subjects: the geometric theory of compact Riemann surfaces of genus greater than one, and the relationship of the Laplace operator with the geometry of such surfaces.The first part of the book is written in textbook form at the graduate level, with few requisites other than background in either differential geometry or complex Riemann surface theory. It begins with an account of the Fenchel-Nielsen approach to Teichmüller Space. Hyperbolic trigonometry and Bers' partition theorem (with a new proof which yields explicit bounds) are shown to be simple but powerful tools in this context. The second part of the book is a self-contained introduction to the spectrum of the Laplacian based on head equations. The approach chosen yields a simple proof that compact Riemann surfaces have the same eigenvalues if and only if they have the same length spectrum. Later chapters deal with recent developments on isospectrality, Sunada's construction, a simplified proof of Wolpert's theorem, and an estimate fo the number of pairwise isospectral non-isometric examples which depends only on genus.Research workers and graduate students interested in compact Riemann surfaces will find here a number of useful tools and insights to apply to their investigations.

Dalla quarta di copertina

This classic monograph is a self-contained introduction to the geometry of Riemann surfaces of constant curvature –1 and their length and eigenvalue spectra. It focuses on two subjects: the geometric theory of compact Riemann surfaces of genus greater than one, and the relationship of the Laplace operator with the geometry of such surfaces. The first part of the book is written in textbook form at the graduate level, with only minimal requisites in either differential geometry or complex Riemann surface theory. The second part of the book is a self-contained introduction to the spectrum of the Laplacian based on the heat equation. Later chapters deal with recent developments on isospectrality, Sunada’s construction, a simplified proof of Wolpert’s theorem, and an estimate of the number of pairwise isospectral non-isometric examples which depends only on genus. Researchers and graduate students interested in compact Riemann surfaces will find this book a useful reference. Anyone familiar with the author's hands-on approach to Riemann surfaces will be gratified by both the breadth and the depth of the topics considered here. The exposition is also extremely clear and thorough. Anyone not familiar with the author's approach is in for a real treat. ― Mathematical Reviews This is a thick and leisurely book which will repay repeated study with many pleasant hours – both for the beginner and the expert. It is fortunately more or less self-contained, which makes it easy to read, and it leads one from essential mathematics to the “state of the art” in the theory of the Laplace–Beltrami operator on compact Riemann surfaces. Although it is not encyclopedic, it is so rich in information and ideas … the reader will be grateful for what has been included in this very satisfying book.―Bulletin of the AMS The book is very well written and quite accessible; there is an excellent bibliography at the end. ―Zentralblatt MATH