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Journal für die reine und angewandte Mathematik

Managing Editor: Weissauer, Rainer

Ed. by Colding, Tobias / Cuntz, Joachim / Huybrechts, Daniel / Hwang, Jun-Muk

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On stable minimal disks in manifolds with nonnegative isotropic curvature

Jingyi Chen1 / Ailana Fraser2

1Department of Mathematics, The University of British Columbia, Vancouver, B.C. V6T 1Z2, Canada. e-mail:

2Department of Mathematics, The University of British Columbia, Vancouver, B.C. V6T 1Z2, Canada. e-mail:

Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal). Volume 2010, Issue 643, Pages 21–37, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102, DOI: 10.1515/crelle.2010.043, June 2010

Publication History

Received:
2007-11-14
Published Online:
2010-06-21

Abstract

Let N be a compact domain with weakly two-convex boundary ∂N in a Riemannian 4-manifold M with nonnegative isotropic curvature. If D is a stable minimal disk in N with ∂D∂N that solves the free boundary problem, then D is infinitesimally holomorphic; moreover, it is ± holomorphic if M is a Kähler surface with positive scalar curvature, and it is holomorphic for some complex structure if M is a hyperkähler surface. We also show that if N is a compact domain in M of dim M ≧ 4 with nonnegative isotropic curvature and ∂N is two-convex, then π 1(∂N) → π 1(N) is injective.

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