Bryant, RL, Some remarks on the geometry of austere manifolds,
Boletim Da Sociedade Brasileira De Matematica, vol. 21 no. 2
(September, 1991),
pp. 133-157, Springer Nature [MR92k:53112], [dvi], [doi]
(last updated on 2025/06/14)
Author's Comments:
An austere submanifold of Euclidean space is one
such that each
of the quadratic forms in the second fundamental form
has its eigenvalues
occuring in oppositely signed pairs. In particular, an
austere submanifold
is minimal, but, except in the case of surfaces, austerity
is much more
restrictive than minimality. The term austere was
coined by Harvey
and Lawson in their fundamental paper Calibrated
Geometries and
characterises those submanifolds whose conormal
bundle is special Lagrangian,
and hence absolutely minimizing.
The largest known class of examples of austere
submanifolds are the
complex submanifolds of complex n-space regarded
as real submanifolds of
Euclidean 2n-space.
In the first part of this paper, I classify the possible
second fundamental
forms of 3- and 4-dimensional austere submanifolds of
Euclidean space and
in the remaining parts of the paper, I determine the
generality of the
3-dimensional austere submanifolds corresponding to
each possible type
of second fundamental form.
The classification of the possible austere second
fundamental forms
in higher dimensions is still unknown and it is also
unknown whether or
not there exist austere 4-manifolds corresponding to
each of the possible
algebraic types of austere second fundamental forms
found in the first
part of the paper. For further progress in the analysis of
some examples
of austere submanifolds, consult the work of Dajczer
and Gromoll,
The
Weierstrass representation for complete minimal real
Kähler submanifolds
of codimension two, Inventiones Mathematicae
119 (1995), 235–242.
Abstract:
We prove several structure theorems about the special class of minimal submanifolds which Harvey and Lawson have called "austere" and which arose in connection with their foundational work on calibrations. The condition of austerity is a pontwise condition on the second fundamental form and essentially requires that the non-zero eigenvalues of the second fundamental form in any normal direction at any point occur in oppositely signed pairs. We solve the pointwise problem of describing the set of austere second fundamental forms in dimension at most four and the local problem of describing the austere three-folds in Euclidean space in all dimensions. © 1991 Sociedade Brasileira de Matemática.
Mathematics Department
