ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

5. Differential Geometry and Global Analysis

1991 MS Classification: 53, 57 Røgen, Peter, Technical University of Denmark: Embedding and knotting of flat surfaces in 3-space In 3-space, any compact surface with nonempty boundary is isotopic to a flat (zero Gaussian curvature) surface and two such flat surfaces are isotopic if and only if they are isotopic through flat surfaces. An akin to Seifert surfaces is though: Any simple closed space curve can be deformed to bound a flat surface. A necessary condition for a link to bound a flat surface with Euler characteristic χ is to have at least 2|χ| 3-singular points, i. e. points with either zero curvature or zero torsion. This is a generalized vertex theorem for closed space curves. A necessary and sufficient condition for a knot to bound an orientable flat surface of a nice, but generic, type will be presented.