Painlevé equations in the differential geometry of surfaces / Alexander I. Bobenko, Ulrich Eitner.
Contributor(s): Eitner, UlrichMaterial type: TextSeries: Lecture notes in mathematics (Springer-Verlag): 1753.Publisher: Berlin ; New York : Springer, ©2000Description: 1 online resource (vi, 120 pages) : illustrationsContent type: text Media type: computer Carrier type: online resourceISBN: 9783540444527; 3540444521Subject(s): Surfaces | Painlevé equations | Painlevé equations | SurfacesGenre/Form: Electronic books. Additional physical formats: Print version:: Painlevé equations in the differential geometry of surfaces.DDC classification: 510 s | 515/.352 LOC classification: QA3 | .L28 no. 1753 | QA649Online resources: Click here to access online
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Includes bibliographical references (pages 113-117) and index.
Basics on painlevé equations and quaternionic description of surfaces -- Bonnet surfaces in euclidean three-space -- Bonnet surface in S³ and H³ and surfaces with harmonic inverse mean curvature -- Surfaces with constant curvature.
This book brings together two different branches of mathematics: the theory of Painlev and the theory of surfaces. Self-contained introductions to both these fields are presented. It is shown how some classical problems in surface theory can be solved using the modern theory of Painlev equations. In particular, an essential part of the book is devoted to Bonnet surfaces, i.e. to surfaces possessing families of isometries preserving the mean curvature function. A global classification of Bonnet surfaces is given using both ingredients of the theory of Painlev equations: the theory of isomonodromic deformation and the Painlev property. The book is illustrated by plots of surfaces. It is intended to be used by mathematicians and graduate students interested in differential geometry and Painlev equations. Researchers working in one of these areas can become familiar with another relevant branch of mathematics.