Transformations and Singularities

Transformations of surfaces or hypersurfaces allow to construct new (families of) surfaces or hypersurfaces of the same kind. For example, an isothermic surface admits a variety of transformations into new isothermic surfaces. This concept of transformations is rather far-reaching: for example, the classical (local) Weierstrass representation of minimal surfaces can be interpreted as a special case of the Goursat transformation for isothermic surfaces. Thus transformations may not just serve to construct new examples or classes of examples of surfaces but may, in some cases, serve to construct all surfaces of a certain class from simple initial data. Singularities are, most generally, the bad points of a theory, that is, points at which the methods of the theory fail. This already constitutes a strong motivation for their study, since it requires to extend the employed methods, to reconsider the objects studied and hence motivates new viewpoints on the theory. Moreover, studying global properties of (hyper-)surfaces, singularities are in many cases inevitable: famous examples are the Caratheodory conjecture, that every (convex) sphere in Euclidean space has at least two umbilics (singularities of its curvature line net), or Hilbert`s theorem on the non-existence of smooth complete pseudospherical surfaces in Euclidean space. The principal aim of this project is to study the interplay between transformations and singularities. More precisely: -- we aim to understand the singularities in the transformation theories of isothermic and Guichard surfaces and of conformally flat hypersurfaces, how their transformations behave (or fail to behave) at certain points; and -- we aim to study how those transformations create (or annihilate) singularities of the transformed (hyper- )surfaces, and what the nature of the occurring singularities is. A good understanding of these two aspects of the interplay between transformations and singularities will ultimately lead to global transformation theories and to natural (global) definitions or characterizations of the studied (hyper-)surface classes. The bilateral collaboration between groups in Japan and in Austria will be key to the success of this research project. In particular, expertise in transformations of surfaces and the integrable systems approach to the relevant classes of (hyper-)surfaces (Austria) and in singularities of surfaces and global surface theory (Japan) will be essential for the project; and exchange of diverse viewpoints on discrete differential geometry in both countries will greatly enrich the related aspects of the project. However, our strategy is to not only bring together experienced experts in the areas relevant to the problems addressed, but to emphasize support of young researchers and PhD students in order to create sustainable and long term research collaborations between research groups in Japan and in Austria.

A transformation transforms a (geometric) object, e.g., a surface or a curve, into a new object of the same kind: thus it can be used to create new surfaces or curves from old ones. On the other hand, transformations can be used to create chains or nets of curves or surfaces --- in this way, for example, a semi-discrete surface can be created as a chain of curves. This project considered certain transformations that relate in particular ways to the geometry of a surface (or curve), such as the Christoffel duality, where the intersection angles of curves on a surface are preserved and the directions of least and greatest curvature on the transformed surface are parallel to those on the original surface. Excluding trivial transformations, e.g., displacements, only special surfaces admit such transformations --- however, the class is large. In geometry, singularities are points, where things go wrong, for example, kinks of a smooth surface or ends of a surface may be considered as singular. A principal aim of this project was to investigate the behaviour of transformations at umbilics, which are points where the aforementioned transformations fail. A better understanding of this behaviour will be paramount to a coherent transformation theory as many interesting geometric configurations force a surface to have such umbilics. The results of this endeavour have been surprising, so far: while the transformations behave reasonably well, as expected, for most of the interesting configurations, they display a wild behaviour at ca 25% of the relevant configurations. This has devastating consequences for a proposed classification of surfaces that admit those transformations, as it shows that a sizeable number of transformations does not respect the class. A key idea in this analysis was to employ methods that we developed during the project in order to understand similar transformations of semi- discrete surfaces, showing how a-priori unrelated topics may cross-fertilize each other. A great strength of our project was its broad views and participation resp researcher base; as a consequence, a large variety of results has been obtained, e.g., also concerning the creation of singularities by transformations. In this context, one topic was the creation and investigation of surfaces in a 3-dimensional spacetime that have lines, where the angle measurement fails, i.e., paths of light speed. Here our investigations relate to an obscure relation of certain transforms with minimal surfaces, i.e., surfaces formed by tensile structures. While the present project was solely aimed at fundamental research in geometry, it is not unlikely that some of our results will have implications for applications, for example, in design: certain semi-discrete surfaces can be used to design physical surfaces out of sheet materials. However, the use of transformations in design processes is a topic of another project.