Flat surfaces in 3-space and their boundaries.

    Project Details

    Description

    A surface in 3-space is called flat if its Gaussian curvature is identically equal to zero.
    The isotopy classes of flat compact surfaces with non-vanishing boundary have, in this project, been proven to be in one-one correspondence with the isotopy classes of ordinary compact surfaces with non-vanishing boundary in 3-space. The exact statement is: In 3-space, any compact surface with non-vanishing boundary is isotopic to a flat surface and two such flat surfaces are isotopic through ordinary surfaces if and only if they are isotopic through flat surfaces.
    Some necessary conditions and one sufficient condition for a knot or link in 3-space to bound a flat surface are found. Most of the obtained results are analogous to recent results on positive curvature surfaces and their boundaries obtained by H. Gluck, L.-H. Pan, and M. Ghomi.
    Long sight goals of this project are to give a necessary and sufficient condition for a knot or link to bound a flat surface and to determine if analogous results holds for negative curvature surfaces in 3-space.
    StatusFinished
    Effective start/end date01/03/199731/01/1999