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

Martin Man-chun Li Chinese University of Hong Kong

Differential Geometry mathscidoc:1910.43018

Proceedings of the American Mathematical Society, 140, (8), 2843-2854, 2012
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.
No keywords uploaded!
[ Download ] [ 2019-10-20 10:58:56 uploaded by Martin_Man_chun_Li ] [ 495 downloads ] [ 0 comments ]
@inproceedings{martin2012on,
  title={On complete stable minimal surfaces in 4-manifolds with positive isotropic curvature},
  author={Martin Man-chun Li},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191020105856720084547},
  booktitle={Proceedings of the American Mathematical Society},
  volume={140},
  number={8},
  pages={2843-2854},
  year={2012},
}
Martin Man-chun Li. On complete stable minimal surfaces in 4-manifolds with positive isotropic curvature. 2012. Vol. 140. In Proceedings of the American Mathematical Society. pp.2843-2854. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191020105856720084547.
Please log in for comment!
 
 
Contact us: office-iccm@tsinghua.edu.cn | Copyright Reserved