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Kotaro Yamada

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Weierstrass Type Representation Formula and its Applications

Kotaro Yamada
Degree: Dr. of Science (Keio University)
Research Interests: Differential Geometry
Unit: Space & Flow

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Our research subjects are classes of surfaces which admits a certain kind of representation formula, which is closely related to a fundamental theory for a modern visualization of curves and surfaces.

The classical Weierstrass representation formula is a explicit expression of minimal surfaces in the Euclidean 3-space--- a mathematical model of soap films, that is, surfaces of the least area among those with given boundaries --- in terms of complex analytic data. Such a representation formula stands over a special choice of parametrization of surfaces. In other words, to express fine representation of surfaces, one need to take a good coordinates.

There are lot of classes of surfaces which have an analogue of the Weierstrass representation under suitable choices of parameters: constant mean curvature surfaces, surfaces of constant mean curvature one in the hyperbolic 3-space, flat surfaces in the hyperbolic 3-spaces, maximal surfaces in the Lorentz-Minkowski spacetime and so on. Our research group has been investigated such classes of surfaces using the Weierstrass-type representation formulae and observed a large amount of properties of them.

At the same time, other members of our GCOE, Tim Hoffmann and Wayne Rossman, are working on discretization of surfaces (for details, please see their respective articles). This is an essential issue to visualize a smooth surface using a computer, and it can be said as “good discretization is born out of a good parameterization”. In fact, Hoffmann has found good discretization using isothermic parameters for a class of surfaces that allow these parameters. The discretization of a flat surface in a hyperbolic space, which appears in Rossman’s article, can be considered as a research project along this line.

Unlike common perceptions, discretized representations like the ones above are far more complex compared to continuous representations using classical differentiation and integration. For this reason, studying properties of surfaces before discretization is essential. In particular, we aim to conduct a detailed study on differential geometrical properties of surfaces which admit Weierstrass type representation formulae.

Hence, my group has been able to develop a fundamental theory on surfaces with a mean curvature one in a hyperbolic space (CMC-1 surfaces) and flat surfaces. As the next step, Udo Hertrich-Jeromin is proposing a certain discretization method of CMC-1 surfaces in the hyperbolic space, while Wayne Rossman is proposing a discretization method of flat surfaces in the hyperbolic space in conjunction with Tim Hoffmann and our research group. In particular, flat surfaces in the hyperbolic space is known as an instance where singularity naturally occurs (in a continuous case). This fact provides a clue to the question, “What is singularity on a discrete surface?”, which is also addressed in Rossman’s article.

I intend for my group to develop differential geometrical theories for classes of surfaces with a variety of good representation formulae in order to provide a direction for discretization and visualization.

Although differential geometry provides a fundamental theory for visualization, visualization technology nowadays is indispensable for research on differential geometry and even wider range of mathematics. In fact, when a researcher discover a special surface, it is natural to show it by computer image. Moreover, it can be said that “experiments” have become possible in differential geometry by visualization. For this reason, research groups at Berlin Institute of Technology and Massachusetts Institute of Technology have been developing visualization software for surfaces for differential geometry research. On the other hand, in Japan, it appears that the importance of visualization as a research tool has not been well recognized.

Currently, just simply entering a representation formula into a mathematical software, like as Mathematica, does not produce a beautiful image; ingenuity is needed to achieve this type of image. Unfortunately skills for this purpose have yet to be accumulated in Japan. Hence, it is hoped that visualization tools will be actively employed at the undergraduate and graduate levels of education, especially for students who do not have backgrounds of differential geometry, and I have written a textbook, “Introduction to Differential Geometry for Engineers”, which introduces readers to differential geometry as they draw pictures. This book can be a tool in preparing to read rigorous textbooks on differential geometry for curves and surfaces (such as the one written by Shoshichi Kobayashi or the one written by Masaaki Umehara and Kotaro Yamada). I sincerely hope that students not majoring in mathematics will find my book useful.

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