AFFINE BERNSTEIN PROBLEMS AND MONGE-AMPÈRE EQUATIONS
by An-Min Li (Sichuan University, China), Ruiwei Xu (Henan Normal University, China), Udo Simon (Technische Universität Berlin, Germany),
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Fang Jia (Sichuan University, China)
Table of Contents (127k)
Preface (126k)
Chapter 1: Basic Tools (204k)
About the Authors
The authors represent three generations of geometers. U Simon finished his doctoral thesis with K P Grotemeyer at the FU Berlin in 1965, and from his lectures he became interested in global differential geometry. U Simon became a professor of mathematics at TU Berlin in 1970.
A M Li started his studies at Peking University in 1963, but because of the cultural revolution he could not finish his MS before 1982. Following a recommendation of S S Chern, he came as AvH fellow to the TU Berlin in 1986 the first time, and he finished his doctoral thesis there with U Simon, U Pinkall and K Nomizu. A M Li has been a professor of mathematics at Sichuan University since 1986, succesfully guiding research groups since then. A M Li was also the advisor of F Jia (PhD 1997) and R Xu (PhD 2008) at Sichuan University. Both are now professors themselves, F Jia at Sichuan University since 1997, R Xu at Henan Normal University since 2008.
The homepages of our Chinese-German cooperation give some more details, for the momentary project see http://www.math.tu-berlin.de/geometrie/gpspde/
In this monograph, the interplay between geometry and partial differential equations (PDEs) is of particular interest. It gives a selfcontained introduction to research in the last decade concerning global problems in the theory of submanifolds, leading to some types of Monge-Ampère equations.
From the methodical point of view, it introduces the solution of certain Monge-Ampère equations via geometric modeling techniques. Here geometric modeling means the appropriate choice of a normalization and its induced geometry on a hypersurface defined by a local strongly convex global graph. For a better understanding of the modeling techniques, the authors give a selfcontained summary of relative hypersurface theory, they derive important PDEs (e.g. affine spheres, affine maximal surfaces, and the affine constant mean curvature equation). Concerning modeling techniques, emphasis is on carefully structured proofs and exemplary comparisons between different modelings.
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