Discrete differential geometry, Lie sphere geometry, discrete surfaces theory, surface representations

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During this fiscal year, Denis Polly used this grant to conduct research in discrete differential geometry. One project is on discrete constant mean curvature 1 surfaces in hyperbolic 3-space, including also myselfand Denis and Udo Hertrich-Jeromin of Vienna Technical University and Andrew Sageman-Furnas of North Carolina State University. Another project is on linear Weingarten surfaces that are also Lie minimal, including also Masaya Hara and Tomohiro Tada of Kobe University, and Joseph Cho of TU-Vienna.Denis also worked with other researchers at other universities in Japan, such as Masashi Yasumoto and Tokushima University, and has potential projects developing as a result.In the first of the two projects, he succeeded in extending the notion of edge-constraint to associated families of discrete constant mean curvature 1 surfaces in hyperbolic 3-space. This is significant because up until now the notion of edge-constraint has been applied only to surfaces in Euclidean 3-space.In the second project, it has been shown that any minimal or constant mean curvature or affine linear Weingarten surfaces in Euclidean 3-space that is also Lie minimal must be a surface of revolution, and that the situation is slightly more complicated in the case of surfaces in a non-Euclidean spaceform.