Austere Matrices, Austere Submanifolds and Dupin Hypersurfaces

TL;DR

This paper constructs new austere submanifolds and proper Dupin hypersurfaces in spheres, providing counterexamples to a classification question and establishing bounds on austere subspace dimensions.

Contribution

It introduces three families of austere submanifolds with flat normal bundle and finds new irreducible proper Dupin hypersurfaces with multiple principal curvatures, answering an open problem.

Findings

Constructed three families of austere submanifolds in spheres.

Discovered three irreducible proper Dupin hypersurfaces with 5 principal curvatures.

Provided an upper bound for the dimension of austere subspaces.

Abstract

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.