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Stable complete minimal surfaces in hyperkähler manifolds

Published online by Cambridge University Press:  04 December 2007

CLAUDIO AREZZO
CLAUDIO AREZZO
Affiliation:
University of Warwick, United Kingdom, and Università di Genova, Italy. e-mail: [email protected]
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Abstract

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In this paper we prove that an isometric stable minimal immersion of a complete oriented surface into a hyperkähler 4-manifold is holomorphic with respect to an orthogonal complex structure, if it satisfies a Bernstein-type assumption on the Gauss-lift. This result generalizes a theorem of Micallef for minimal surfaces in the euclidean 4-space. An example found by Atiyah and Hitchin shows that the assumption on the Gauss-lift is necessary.

Keywords

minimal surfaceskähler anglehyperkähler manifolds.
Type
Research Article
Information
Compositio Mathematica , Volume 112 , Issue 1 , May 1998 , pp. 33 - 40
Copyright
© 1998 Kluwer Academic Publishers