Tight polyhedral models of isoparametric families, and PL-taut submanifolds

Abstract

We present polyhedral models for isoparametric families in the sphere with at most three principal curvatures. Each member of the family (including the analogues of the focal sets) is tight in the boundary complex of an ambient convex polytope. In particular, the tube around the real (or complex) Veronese surface is represented as a tight polyhedron in 5-space (or 8-space). The examples are based on a certain Bier sphere triangulation of S⁴ or S⁷, respectively. In the 4-dimensional case there are simplicial branched coverings of these triangulations in the complex projective plane and in S² × S² which are branched precisely along the polyhedral analogues of the Veronese surface. Moreover, we introduce a notion of PL-tautness and discuss its relationship with tightness of polyhedra. In particular, each member of our polyhedral isoparametric family is PL-taut. For an extended abstract see [W. Kühnel, Discrete models of isoparametric hypersurfaces in spheres. Oberwolfach Reports12, 665–667 (2006).].