Finite Möbius Groups, Minimal Immersions of Spheres, and Moduli - (Universitext) by Gabor Toth (Hardcover)

Book Synopsis

"Spherical soap bubbles" or spherical immersions, belong to a fast growing and fascinating area between algebra and geometry. This theory has rich interconnections with a variety of mathematical disciplines. In this book, the author traces the development of the study of spherical minimal immersions. In trying to make this monograph accessible not just to research mathematicians but also to graduate students, the author included sizeable pieces of material from upper level undergraduate courses as well as a valuable selection of exercises at the end of each chapter.

From the Back Cover

"Spherical soap bubbles", isometric minimal immersions of round spheres into round spheres, or spherical immersions for short, belong to a fast growing and fascinating area between algebra and geometry. This theory has rich inteconnections with a variety of mathematical disciplines such as invariant theory, convex geometry, harmonic maps, and orthogonal multiplications. In this book, the author traces the development of the study of spherical minimal immersions over the past 30 plus years, including Takahashi's 1966 proof regarding the existence of isometric minimal immersions, DoCarmo and Wallach's study of the uniqueness of the standard minimal immersion in the seventies, and the mor recent study of the variety of spherical minimal immersions which have been obtained by the "equivariant construction" as SU(2)-orbits, first used by Mashimo in 1984 and then later by DeTurck and Ziller in 1992. In trying to make this monograph accessible not just to research mathematicians but mathematics graduate students as well, the author included sizeable pieces of material from upper level undergraduate courses, additional graduate level topics such as Felix Kleins classic treatise of the icosahedron, and a valuable selection of exercises at the end of each chapter.

Review Quotes

From the reviews of the first edition:

"This monograph is devoted to minimal immersions between round spheres. ... Indeed, the author's exposition is largely self-contained - and leisurely. Exercises are abundant, forming an integral part of the presentation. ... This monograph has been written with great care. It would be appropriate for an undergraduate or graduate seminar." (J. Ells, Mathematical Reviews, Issue 2002 i)

"In this book, the author traces the development of the study of spherical minimal immersions over the past 30-plus years ... . In trying to make this monograph accessible not just to research mathematicians but to mathematics graduate students as well, the author included sizeable pieces of material from upper-level undergraduate courses, additional graduate level topics such as Felix Klein's classical treatise of the icosahedron, and a valuable selection of exercises." (L'Enseignement Mathematique, Vol. 48 (1-2), 2002)

"This very interesting monograph for researchers and graduate students as well, gives a wide picture of the theory of spherical soap bubbles, which studies isometric minimal immersions of round spheres ... . Each chapter ends with some additional topics and some challenging problems. A useful appendix with some basic notions is given at the end of the book." (Cornelia-Livia Bejan, Zentralblatt MATH, Vol. 1074, 2005)