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April 2014 The Chern–Osserman inequality for minimal surfaces in a Cartan–Hadamard manifold with strictly negative sectional curvatures
Antonio Esteve, Vicente Palmer
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Ark. Mat. 52(1): 61-92 (April 2014). DOI: 10.1007/s11512-013-0182-3

Abstract

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 KNb<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 KNb<0.

Funding Statement

The two authors were partially supported by the Mineco-FEDER grant MTM2010-21206-C02-02.

Citation

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Antonio Esteve. Vicente Palmer. "The Chern–Osserman inequality for minimal surfaces in a Cartan–Hadamard manifold with strictly negative sectional curvatures." Ark. Mat. 52 (1) 61 - 92, April 2014. https://doi.org/10.1007/s11512-013-0182-3

Information

Received: 11 May 2012; Revised: 29 November 2012; Published: April 2014
First available in Project Euclid: 30 January 2017

zbMATH: 1318.53063
MathSciNet: MR3175294
Digital Object Identifier: 10.1007/s11512-013-0182-3

Rights: 2013 © Institut Mittag-Leffler

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