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**Harmonic maps and minimal immersions with symmetries: methods of ordinary differential equations applied to elliptic variational problems.**
*(English)*
Zbl 0783.58003

Annals of Mathematics Studies. 130. Princeton, NJ: Princeton University Press,. 228 p. (1993).

Harmonic maps and minimal immersions are two of the fundamental geometric objects which are useful in investigating geometric and topological properties of manifolds and have various applications in related branches of mathematics and physics. As solutions to certain elliptic variational problems, their existence cannot be expected in general. But in the presence of suitable symmetry, they often admit reductions to lower dimensional problems – ordinary differential equations in some cases – whose qualitative study is more manageable. Various reduction techniques play an important role in this procedure.

This monograph is an excellent exposition to the study of harmonic maps, minimal and parallel mean curvature immersions in various symmetric contexts. Many important previously known results in this field are included and well organized.

The book is in three parts, subdivided into ten chapters and four appendices. Each chapter has a brief introduction pointing the way and providing basic motivation. Each chapter ends with a section on notes and comments which give indications of further or alternative directions.

The first part gives the basic variational and geometrical properties of the subjects and general reduction techniques. Part two is a study of \(G\)-invariant minimal and constant mean curvature immersions. The last part is dedicated to the harmonic maps between spheres.

This monograph is an excellent exposition to the study of harmonic maps, minimal and parallel mean curvature immersions in various symmetric contexts. Many important previously known results in this field are included and well organized.

The book is in three parts, subdivided into ten chapters and four appendices. Each chapter has a brief introduction pointing the way and providing basic motivation. Each chapter ends with a section on notes and comments which give indications of further or alternative directions.

The first part gives the basic variational and geometrical properties of the subjects and general reduction techniques. Part two is a study of \(G\)-invariant minimal and constant mean curvature immersions. The last part is dedicated to the harmonic maps between spheres.

Reviewer: Pan Yanglian (Shanghai)