The Chern–Osserman inequality for minimal surfaces in a Cartan–Hadamard manifold with strictly negative sectional curvatures

Antonio Esteve Instituto de Enseñanza Secundaria Alfonso VIII Departamento de Análisis Económico y Finanzas, Universidad de Castilla-La Mancha Vicente Palmer Department of Mathematics Institute of New Imaging Technologies, Universitat Jaume I

Differential Geometry mathscidoc:1701.10014

Arkiv for Matematik, 52, (1), 61-92, 2012.5
We state and prove a Chern–Osserman-type inequality in terms of the volume growth for minimal surfaces$S$which have finite total extrinsic curvature and are properly immersed in a Cartan–Hadamard manifold$N$with sectional curvatures bounded from above by a negative quantity$K$_{$N$}≤$b$<0 and such that they are not too curved (on average) with respect to the hyperbolic space with constant sectional curvature given by the upper bound$b$. We also prove the same Chern–Osserman-type inequality for minimal surfaces with finite total extrinsic curvature and properly immersed in an asymptotically hyperbolic Cartan–Hadamard manifold$N$with sectional curvatures bounded from above by a negative quantity$K$_{$N$}≤$b$<0.
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@inproceedings{antonio2012the,
  title={The Chern–Osserman inequality for minimal surfaces in a Cartan–Hadamard manifold with strictly negative sectional curvatures},
  author={Antonio Esteve, and Vicente Palmer},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203636888705055},
  booktitle={Arkiv for Matematik},
  volume={52},
  number={1},
  pages={61-92},
  year={2012},
}
Antonio Esteve, and Vicente Palmer. The Chern–Osserman inequality for minimal surfaces in a Cartan–Hadamard manifold with strictly negative sectional curvatures. 2012. Vol. 52. In Arkiv for Matematik. pp.61-92. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203636888705055.
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