The Calabi-Yau property of superminimal surfaces in self-dual Einstein four-manifolds

Abstract

In this paper, we show that if $(X,g)$ is an oriented four dimensional Einstein manifold which is self-dual or anti-self-dual then superminimal surfaces in $X$ of appropriate spin enjoy the Calabi-Yau property, meaning that every immersed surface of this type from a bordered Riemann surface can be uniformly approximated by complete superminimal surfaces with Jordan boundaries. The proof uses the theory of twistor spaces and the Calabi-Yau property of holomorphic Legendrian curves in complex contact manifolds.

arXiv commentJ. Geom. Anal., to appear

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Cited by in corpus (5)

• Recent developments on Oka manifolds
• Mergelyan approximation theorem for holomorphic Legendrian curves
• Holomorphic Legendrian curves in ${\mathbb{CP}}^3$ and superminimal surfaces in ${\mathbb{S}}^4$
• Proper superminimal surfaces of given conformal types in the hyperbolic four-space
• Minimal surfaces with symmetries