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.

中文翻译：

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自对偶爱因斯坦四流形上极小表面的卡拉比丘性质

在本文中，我们证明，如果（X，  g）是有向的四维爱因斯坦流形，它是自对偶的或反自对偶的，则适当自旋的X中的极小曲面将具有Calabi–Yau性质，这意味着带有边界的Riemann表面的这种类型的浸入表面可以通过具有Jordan边界的完全极小表面均匀地近似。该证明使用了扭曲空间理论和复杂接触流形中全纯Legendrian曲线的Calabi–Yau性质。