Two-dimensional geometric variational problems.

*(English)*Zbl 0729.49001
Pure and Applied Mathematics. Chichester etc.: John Wiley & Sons. x, 236 p. £35.00 (1991).

This book is devoted to the study of variational problems for mappings from a surface equipped with a conformal structure into Euclidean space or a Riemannian manifold, specifically those problems which are invariant under conformal reparametrizations of the domain. Such problems have gained great significance because J. Douglas’ celebrated solution to the Plateau problem is a special case. Indeed, the author presents the solution of the Plateau problem in the first six pages of the first chapter as the prototypical example. The level of sophistication expected of the reader seems to be fairly high.

The general variational problem covered in this book includes as special cases conformal mappings between surfaces, minimal surfaces in Riemannian manifolds, harmonic maps from a surface into a Riemannian manifold, and solutions of prescribed mean curvature equations.

The book addresses the general theory of existence and regularity for these problems. The chapter of regularity is in part expository, and makes use of extensive references to the author’s earlier book, “Harmonic mappings between Riemannian manifolds” (1983; Zbl 0542.58001).

Further chapters provide applications of the general theory to conformal representation, harmonic maps between surfaces, and Teichmüller theory. The approach to Teichmüller theory is new, relying on harmonic maps rather than the usual quasiconformal maps.

The general variational problem covered in this book includes as special cases conformal mappings between surfaces, minimal surfaces in Riemannian manifolds, harmonic maps from a surface into a Riemannian manifold, and solutions of prescribed mean curvature equations.

The book addresses the general theory of existence and regularity for these problems. The chapter of regularity is in part expository, and makes use of extensive references to the author’s earlier book, “Harmonic mappings between Riemannian manifolds” (1983; Zbl 0542.58001).

Further chapters provide applications of the general theory to conformal representation, harmonic maps between surfaces, and Teichmüller theory. The approach to Teichmüller theory is new, relying on harmonic maps rather than the usual quasiconformal maps.

Reviewer: H.Parks (Corvallis)

##### MSC:

49-02 | Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control |

49Q05 | Minimal surfaces and optimization |

58J60 | Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.) |

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

53A30 | Conformal differential geometry (MSC2010) |

53A10 | Minimal surfaces in differential geometry, surfaces with prescribed mean curvature |