# A new class of austere submanifolds

## TL;DR

This paper introduces a new family of explicit austere submanifolds in Euclidean space that are non-Kaehler and of higher rank, expanding the known examples beyond minimal Kaehler cases.

## Contribution

It provides the first explicit construction of higher-rank, non-Kaehler austere submanifolds using holomorphic data and a Weierstrass type parametrization.

## Key findings

- First explicit examples of higher-rank, non-Kaehler austere submanifolds
- Construction method based on holomorphic data
- Uses a Weierstrass type parametrization

## Abstract

Austere submanifolds of Euclidean space were introduced in 1982 by Harvey and Lawson in their foundational work on calibrated geometries. In general, the austerity condition is much stronger than minimality since it express that the nonzero eigenvalues of the shape operator of the submanifold appear in opposite pairs for any normal vector at any point. Thereafter, the challenging task of finding non-trivial explicit examples, other than minimal immersions of Kaehler manifolds, only turned out submanifolds of rank two, and these are of limited interest in the sense that in this special situation austerity is equivalent to minimality. In this paper, we present the first explicitly given family of austere non-Kaehler submanifolds of higher rank, and these are produced from holomorphic data by means of a Weierstrass type parametrization.