| 研究開始時の研究の概要 |
Edge-constraint conditions for constant mean curvature 1 surfaces will be investigated using established Weierstrass-type representations, followed by DPW-type representations. Once this is accomplished we will aim for applying edge-constraint conditions to more general non-constant mean curvature surface theory as well. Ample use of integrable systems and transformation theory will be involved in the research. Notions of discrete curvature that apply to more general ambient spaces, including their quotient spaces, will be established.
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| 研究実績の概要 |
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.
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