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On complete stable minimal surfaces in 4-manifolds with positive isotropic curvature

Author: Martin Man-chun Li
Journal: Proc. Amer. Math. Soc. 140 (2012), 2843-2854
MSC (2010): Primary 53A10; Secondary 32Q10
Published electronically: November 29, 2011
MathSciNet review: 2910771
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Abstract: We prove the nonexistence of stable immersed minimal surfaces uniformly conformally equivalent to $ \mathbb{C}$ in any complete orientable four-
dimensional Riemannian manifold with uniformly positive isotropic curvature. We also generalize the same nonexistence result to higher dimensions provided that the ambient manifold has uniformly positive complex sectional curvature. The proof consists of two parts: assuming an ``eigenvalue condition'' on the $ \overline {\partial }$-operator of a holomorphic bundle, we prove (1) a vanishing theorem for these holomorphic bundles on $ \mathbb{C}$ and (2) an existence theorem for holomorphic sections with controlled growth by Hörmander's weighted $ L^2$-method.

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Additional Information

Martin Man-chun Li
Affiliation: Department of Mathematics, Stanford University, Stanford, California 94305
Address at time of publication: Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver, BC V6T 1Z2, Canada

Received by editor(s): November 5, 2010
Received by editor(s) in revised form: March 4, 2011
Published electronically: November 29, 2011
Communicated by: Jianguo Cao
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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