Austere Submanifolds of Dimension Four: Examples and Maximal Types

TL;DR

This paper classifies 4-dimensional austere submanifolds in Euclidean space, linking type A to Kahler submanifolds, constructing new examples, and analyzing maximal second fundamental forms for types B and C.

Contribution

It establishes a correspondence between type A austere submanifolds and Kahler geometry, provides new explicit examples, and classifies maximal forms for types B and C.

Findings

Type A submanifolds are exactly real Kahler submanifolds.

New examples of austere submanifolds in R^6 and R^10 are constructed.

Classification results for types B and C with maximal second fundamental forms.

Abstract

Austere submanifolds in Euclidean space were introduced by Harvey and Lawson in connection with their study of calibrated geometries. The algebraic possibilities for second fundamental forms of 4-dimensional austere submanifolds were classified by Bryant, into three types which we label A, B, and C. In this paper, we show that type A submanifolds correspond exactly to real Kahler submanifolds, we construct new examples of such submanifolds in R^6 and R^10, and we obtain classification results on submanifolds of types B and C with maximal second fundamental forms.