Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Franc Forstneric

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.

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

Abstract

In this paper, we show that if is an oriented four dimensional Einstein manifold which is self-dual or anti-self-dual then superminimal surfaces in 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.

Paper Structure

This paper contains 6 sections, 14 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. Superminimal surfaces in Riemannian four-manifolds
  3. Almost hermitian structures on $\mathbb{R}^4$ and quaternions
  4. Twistor bundles and the Bryant correspondence
  5. Proofs of Theorems \ref{['th:main']} and \ref{['th:main2']}
  6. Twistor spaces of the $4$-sphere and of the hyperbolic $4$-space

Key Result

Theorem 1.2

Let $(X,g)$ be an oriented four dimensional Einstein manifold whose Weyl tensor $W=W^++W^-$ satisfies $W^+=0$ or $W^-=0$. Given any bordered Riemann surface $M$ and a conformal superminimal immersion $f_0\in \mathrm{SM}^\pm(\overline M,X)$ of class $\mathscr{C}^3$ (with the respective choice of sign

Theorems & Definitions (33)

  • Definition 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Definition 2.1
  • Remark 2.2
  • Lemma 3.1
  • proof
  • Proposition 4.1
  • Example 4.2
  • Remark 4.3
  • ...and 23 more