Sfb 288 Differential Geometry and Quantum Physics

Abstract for Sfb Preprint No. 198

On Superminimal Surfaces

Th. Friedrich

Using the Cartan method O. Boruvka (see [B1], [B2]) studied superminimal surfaces in four-dimensional space forms. In particular, he described locally the family of all superminimal surfaces and classified all of them with a constant radius of the indicatrix. We discuss the mentioned results from the point of view of the twistor theory, providing some new proofs. It turns out that the superminimal surfaces investigated by geo-meters at the beginning of this century as well as by O. Boruvka have a holomorphic and horizontal lift into the twistor space. Global results concerning superminimal surfaces have been obtained during the last 15 years. In this paper we investigate superminimal surfaces in the hyperbolic four-spaces.\

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