A new class of austere submanifolds

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 expresses 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 the geometrically simple generalized helicoids and the submanifolds of rank two. In fact, the latter are of limited interest in the sense that austerity is equivalent to minimality in this special situation. In the present paper, we present the first explicitly given family of austere non-Kaehler submanifolds of higher rank, other than the generalized helicoids, and these are produced from holomorphic data by means of a Weierstrass type parametrization.

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Received 25 November 2019

Accepted 3 February 2021

Published 16 July 2024