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 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 -operator of a holomorphic bundle, we prove (1) a vanishing theorem for these holomorphic bundles on and (2) an existence theorem for holomorphic sections with controlled growth by Hörmander's weighted -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

Email:
martinli@stanford.edu, martinli@math.ubc.ca

DOI:
http://dx.doi.org/10.1090/S0002-9939-2011-11113-4

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.