##
**Lectures on differential geometry.**
*(English)*
Zbl 0830.53001

Conference Proceedings and Lecture Notes in Geometry and Topology. 1. Cambridge, MA: International Press. v, 414 p. (1994).

This book is an excellent exposition of differential geometry from an analytic point of view. This aspect of differential geometry is a very active field of research nowadays. In particular the work of the authors had a great impact on this development. The main topics of the first six chapters are eigenvalue problems for the Laplacian, comparison results, harmonic functions on manifolds of negative curvature, heat kernel estimates, the Yamabe problem and locally conformally flat manifolds.

The book consists of nine chapters. The first four chapters contain a series of lectures the authors gave at the Institute for Advanced Studies in Princeton in 1984. In the academic year 1984/85 the authors gave lectures in San Diego which in part are the content of chapters V and VI. The first version of the book was written in Chinese, it was translated into English by S. Y. Cheng and W. Y. Ding. The authors added some updating of the material. Here one has to note that – in particular due to the work and the influence of the authors – there has been much progress in this field in the last years. The last three chapters are reprints of articles by S. T. Yau originally published in other journals resp. books. Chapter VII is a reprint of the famous Problem section of the Seminar in Differential Geometry at Tokyo [Ann. Math. Stud. 102, 669-706 (1982; Zbl 0479.53001)], Chapter IX is a reprint of the problem list “Open problems in geometry” [Proc. Symp. Pure Math. 54, Part 1, 1-28 (1993; Zbl 0801.53001)]. The article “Nonlinear analysis in geometry” [Monographie de l’Enseignement Mathématique. 33, Université de Genève. 54 p. (1986; Zbl 0631.53003)] is included as Chapter VIII, being an extended version of lectures S. T. Yau gave at the ETH Zürich in 1981.

In the first chapter comparison results for the Laplace operator are given. For example, the volume growth on manifolds of non-negative Ricci curvature is investigated. Chapter II deals with harmonic functions on manifolds with negative curvature. For instance, a simple proof of the following result by Anderson and Sullivan is given: There exists a bounded harmonic function on a complete manifold whose sectional curvature is pinched between two negative constants. The topic of chapter III are eigenvalue problems of the Laplace operator. At first Cheeger’s inequality is presented, then several results on lower bounds of the first eigenvalue of the Laplace operator due to Li and Yau are discussed. As a tool gradient estimates for the first eigenfunction are used. Estimates for higher eigenvalues and for spectral gaps are also given. Finally estimates on surfaces extending Hersch’s upper bound for \(\lambda_1\) on the 2-sphere in terms of the area are given for higher genus surfaces. Chapter IV deals with the heat kernel on Riemannian manifolds. Applying estimates for the heat kernel, one obtains estimates for the spectrum of a compact Riemannian manifold. The topic of chapter V is the Yamabe problem, which was finally solved by T. Aubin and R. Schoen. The proof follows the approach given by J. Lee and T. Parker which is based on the introduction of conformal normal coordinates and the expansion of the Green’s function in these coordinates. Chapter VI deals with locally conformally flat manifolds. Here in particular Kleinian manifolds are studied. They are of the form \(\Omega/\Gamma\) where \(\Gamma\) is a Kleinian group and \(\Omega\) is the domain of discontinuity of \(\Gamma\). For example, the authors give a necessary and sufficient condition for the existence of a metric of non-negative scalar curvature in the conformal class of a Kleinian manifold \(\Omega/ \Gamma\) in terms of the dimension of the limit set.

The book consists of nine chapters. The first four chapters contain a series of lectures the authors gave at the Institute for Advanced Studies in Princeton in 1984. In the academic year 1984/85 the authors gave lectures in San Diego which in part are the content of chapters V and VI. The first version of the book was written in Chinese, it was translated into English by S. Y. Cheng and W. Y. Ding. The authors added some updating of the material. Here one has to note that – in particular due to the work and the influence of the authors – there has been much progress in this field in the last years. The last three chapters are reprints of articles by S. T. Yau originally published in other journals resp. books. Chapter VII is a reprint of the famous Problem section of the Seminar in Differential Geometry at Tokyo [Ann. Math. Stud. 102, 669-706 (1982; Zbl 0479.53001)], Chapter IX is a reprint of the problem list “Open problems in geometry” [Proc. Symp. Pure Math. 54, Part 1, 1-28 (1993; Zbl 0801.53001)]. The article “Nonlinear analysis in geometry” [Monographie de l’Enseignement Mathématique. 33, Université de Genève. 54 p. (1986; Zbl 0631.53003)] is included as Chapter VIII, being an extended version of lectures S. T. Yau gave at the ETH Zürich in 1981.

In the first chapter comparison results for the Laplace operator are given. For example, the volume growth on manifolds of non-negative Ricci curvature is investigated. Chapter II deals with harmonic functions on manifolds with negative curvature. For instance, a simple proof of the following result by Anderson and Sullivan is given: There exists a bounded harmonic function on a complete manifold whose sectional curvature is pinched between two negative constants. The topic of chapter III are eigenvalue problems of the Laplace operator. At first Cheeger’s inequality is presented, then several results on lower bounds of the first eigenvalue of the Laplace operator due to Li and Yau are discussed. As a tool gradient estimates for the first eigenfunction are used. Estimates for higher eigenvalues and for spectral gaps are also given. Finally estimates on surfaces extending Hersch’s upper bound for \(\lambda_1\) on the 2-sphere in terms of the area are given for higher genus surfaces. Chapter IV deals with the heat kernel on Riemannian manifolds. Applying estimates for the heat kernel, one obtains estimates for the spectrum of a compact Riemannian manifold. The topic of chapter V is the Yamabe problem, which was finally solved by T. Aubin and R. Schoen. The proof follows the approach given by J. Lee and T. Parker which is based on the introduction of conformal normal coordinates and the expansion of the Green’s function in these coordinates. Chapter VI deals with locally conformally flat manifolds. Here in particular Kleinian manifolds are studied. They are of the form \(\Omega/\Gamma\) where \(\Gamma\) is a Kleinian group and \(\Omega\) is the domain of discontinuity of \(\Gamma\). For example, the authors give a necessary and sufficient condition for the existence of a metric of non-negative scalar curvature in the conformal class of a Kleinian manifold \(\Omega/ \Gamma\) in terms of the dimension of the limit set.

Reviewer: H.-B.Rademacher (Leipzig)

### MSC:

53-02 | Research exposition (monographs, survey articles) pertaining to differential geometry |

58-02 | Research exposition (monographs, survey articles) pertaining to global analysis |

53C21 | Methods of global Riemannian geometry, including PDE methods; curvature restrictions |

58C40 | Spectral theory; eigenvalue problems on manifolds |

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

53C20 | Global Riemannian geometry, including pinching |