Geometry and spectra of compact Riemann surfaces.

*(English)*Zbl 0770.53001
Progress in Mathematics (Boston, Mass.). 106. Boston, MA: Birkhäuser. xiv, 454 p. (1992).

The book deals with two main subjects. The first is the geometric theory of compact Riemann surfaces of genus greater than one; the second is the geometry of the Laplace operator on compact Riemann surfaces.

The first six chapters provide a lovely introduction to the geometry of compact Riemann surfaces based on hyperbolic geometry and on cutting and pasting. Chapter 1 deals with properties of surfaces obtained by pasting geodesic polygons from the hyperbolic plane, Chapter 2 discusses hyperbolic trigonometry. Chapters 3 and 6 give the construction of compact Riemann surfaces and lead to the Frenchel-Nielsen model of Teichmüller space. Chapters 4 and 5 contain the basic qualitative geometric results about Riemann surfaces: the collar theorem and the Bers’ theorem on length controlled pants decomposition.

The second part of the book contains an introduction to the spectrum of the Laplace operator based on the heat kernel. Chapter 8 deals with small eigenvalues, Chapter 9 with closed geodesics and Huber’s theorem. Chapter 10 deals with Wolpert’s theorem. Chapter 11 deals with Sunada’s theorem. Chapter 12 provides examples of isospectral non isometric Riemann surfaces. Chapter 13 discusses the size of isospectral families. Chapter 14 discusses perturbation theory.

The book is very well written and quite accessible; there is an excellent bibliography at the end.

The first six chapters provide a lovely introduction to the geometry of compact Riemann surfaces based on hyperbolic geometry and on cutting and pasting. Chapter 1 deals with properties of surfaces obtained by pasting geodesic polygons from the hyperbolic plane, Chapter 2 discusses hyperbolic trigonometry. Chapters 3 and 6 give the construction of compact Riemann surfaces and lead to the Frenchel-Nielsen model of Teichmüller space. Chapters 4 and 5 contain the basic qualitative geometric results about Riemann surfaces: the collar theorem and the Bers’ theorem on length controlled pants decomposition.

The second part of the book contains an introduction to the spectrum of the Laplace operator based on the heat kernel. Chapter 8 deals with small eigenvalues, Chapter 9 with closed geodesics and Huber’s theorem. Chapter 10 deals with Wolpert’s theorem. Chapter 11 deals with Sunada’s theorem. Chapter 12 provides examples of isospectral non isometric Riemann surfaces. Chapter 13 discusses the size of isospectral families. Chapter 14 discusses perturbation theory.

The book is very well written and quite accessible; there is an excellent bibliography at the end.

Reviewer: P.Gilkey (Eugene)

##### MSC:

53-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to differential geometry |

32G15 | Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) |

30F20 | Classification theory of Riemann surfaces |

53A35 | Non-Euclidean differential geometry |

58J50 | Spectral problems; spectral geometry; scattering theory on manifolds |