Austere Matrices, Austere Submanifolds and Dupin Hypersurfaces

Motivated by Bryant's research on austere subspaces and Cartan's isoparametric hypersurfaces with 3 distinct principal curvatures, we construct three families of austere submanifolds with flat normal bundle in unit spheres. From these examples we find three irreducible proper Dupin hypersurfaces with 5 distinct principal curvatures of different multiplicities. Thus, we give a negative answer to an open question raised by Thorbergsson in 2000 which is instructive for the local classification of proper Dupin hypersurfaces. Moreover, as an application, we obtain an upper bound estimate for the dimension of austere subspaces.