Problems in Lie Sphere Geometry

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The goal of this proposal is to prove that any irreducible proper Dupin hypersurface with four principal curvatures whose cross ratio is constant is equivalent by a Lie sphere transformation to an isoparametric hypersurface. The methods will build on the successful solution of this problem in the case of three principal curvatures. As in that case, the first step will be to prove that the multiplicities of the principal curvatures satisfy the relations required by an isoparametric hypersurface. A new feature when there are more than three principal curvatures is the cross ratio of any four principal curvatures. The next step is to prove that if this cross ratio is constant, then it must equal its value in the isoparametric case. Given the correct conditions on the multiplicities and the cross ratio, the final step is to prove that irreducibility implies that the hypersurface is equivalent by Lie sphere transformations to an isoparametric hypersurface. In each case, the method of moving frames will be used to study the local differential geometry of Dupin hypersurfaces in Lie sphere geometry.A surface in ordinary space is called isoparametric if its principal curvatures are constant. This condition is so restrictive that the only examples are planes, spheres, and circular cylinders. If each principal curvature is assumed constant only along each of its lines of curvature, then the surface is called a cyclide of Dupin. Cyclides are used by mechanical engineers because of the way such surfaces can be fit together along their circles of curvature. A cyclide that is not isoparametric is the torus of revolution, which is the surface of an ordinary doughnut. Any cyclide with two distinct principal curvatures can be obtained from a single torus by rigid motions, inversions in spheres, or parallel translations (the locus of points a fixed distance from the surface). All such transformations generate the group of Lie sphere transformations. In more variables, there are many remarkable isoparametric hypersurfaces in spheres, but they can have only 1, 2, 3, 4, or 6 distinct principal curvatures. Dupin's cyclides generalize to the idea of Dupin hypersurfaces in spheres. These include the isoparametric hypersurfaces. There is a local construction that builds a new Dupin hypersurface from one of smaller dimension. A Dupin hypersurface is called irreducible if it is not built by one of these constructions. For an isoparametric hypersurface with four distinct principal curvatures, the cross ratio of the principal curvatures is constant. The goal of this proposal is to prove that, if an irreducible Dupin hypersurface has four distinct principal curvatures with constant cross ratio, then it is a Lie sphere transformation of an isoparametric hypersurface.