# The Calabi–Yau Property of Superminimal Surfaces in Self-Dual Einstein Four-Manifolds

@article{Forstneri2020TheCP,
title={The Calabi–Yau Property of Superminimal Surfaces in Self-Dual Einstein Four-Manifolds},
author={F. Forstneri{\vc}},
journal={The Journal of Geometric Analysis},
year={2020},
pages={1-27}
}
• F. Forstnerič
• Published 2020
• Mathematics
• The Journal of Geometric Analysis
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… Expand
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#### References

SHOWING 1-10 OF 88 REFERENCES
Second variation of superminimal surfaces into self-dual Einstein four-manifolds
• Mathematics
• 1997
The index of a compact orientable superminimal surface of a selfdual Einstein four-manifold M with positive scalar curvature is computed in terms of its genus and area. Also a lower bound of itsExpand
Toric Anti-self-dual 4-manifolds Via Complex Geometry
• Mathematics
• 2006
Using the twistor correspondence, this article gives a one-to-one correspondence between germs of toric anti-self-dual conformal classes and certain holomorphic data determined by the induced actionExpand
Toric anti-self-dual Einstein metrics via complex geometry
Using the twistor correspondence, we give a classification of toric anti-self-dual Einstein metrics: each such metric is essentially determined by an odd holomorphic function. This explains how theExpand
The Riemannian geometry of superminimal surfaces in complex space forms
This paper deals with superminimal surfaces in complex space forms by using the Frenet framing. We formulate explicitly the length squares of the higher fundamental forms and the higher curvaturesExpand
On The Ricci Curvature of a Compact Kahler Manifold and the Complex Monge-Ampere Equation, I*
Therefore a necessary condition for a (1,l) form ( G I a ' r r ) I,,, Rlr dz' A d? to be the Ricci form of some Kahler metric is that it must be closed and its cohomology class must represent theExpand
On superminimal surfaces
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 andExpand
FANO MANIFOLDS, CONTACT STRUCTURES, AND QUATERNIONIC GEOMETRY
Let Z be a compact complex (2n+1)-manifold which carries a complex contact structure, meaning a codimension-1 holomorphic sub-bundle D⊂TZ which is maximally non-integrable. If Z admits aExpand
HOLOMORPHIC LEGENDRIAN CURVES IN CP AND SUPERMINIMAL SURFACES IN S4
We obtain a Runge approximation theorem for holomorphic Legendrian curves and immersions in the complex projective 3-space CP, both from open and compact Riemann surfaces, and we prove that the spaceExpand
Higher Singularities and the Twistor Fibration π: CP3 → S4
• Mathematics
• 2000
We use the Klein correspondences to write down an explicit relationship between two holomorphic curves, namely the directrix curve and the twistor lift, associated to a superminimal map from aExpand
Métriques autoduales sur la boule
Abstract.A conformal metric on a 4-ball induces on the boundary 3-sphere a conformal metric and a trace-free second fundamental form. Conversely, such a data on the 3-sphere is the boundary of aExpand