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Harmonic univalent mappings and minimal graphs. (English) Zbl 1316.31002

Joshi, Santosh (ed.) et al., Current topics in pure and computational complex analysis. Most of the selected papers based on the presentations at the international workshop on complex analysis and its applications, Sangli, India, June 11–15, 2012. New Delhi: Birkhäuser/Springer (ISBN 978-81-322-2112-8/hbk; 978-81-322-2113-5/ebook). Trends in Mathematics, 21-46 (2014).
Summary: We survey results and open problems in harmonic maps and minimal surface theory at a level appropriate for graduate students and others interested in contributing to the existing research. After covering some basic results, several topics are covered in more detail, including the shearing technique, inner mapping radius, convolutions, the Weierstrass Representation, determining minimal surfaces via change of variables, curvature bounds, and conjugate minimal surfaces. A variety of new and standing conjectures is included throughout. Examples are worked in detail and presented visually using ComplexTool, Mathematica, and other software packages.
For the entire collection see [Zbl 1305.30003].

MSC:

31A05 Harmonic, subharmonic, superharmonic functions in two dimensions
49Q05 Minimal surfaces and optimization
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