## Arkiv för Matematik

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

#### 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.

#### Note

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

#### Article information

Source
Ark. Mat., Volume 52, Number 1 (2014), 61-92.

Dates
Revised: 29 November 2012
First available in Project Euclid: 30 January 2017

https://projecteuclid.org/euclid.afm/1485802656

Digital Object Identifier
doi:10.1007/s11512-013-0182-3

Mathematical Reviews number (MathSciNet)
MR3175294

Zentralblatt MATH identifier
1318.53063

Rights

#### Citation

Esteve, Antonio; Palmer, Vicente. The Chern–Osserman inequality for minimal surfaces in a Cartan–Hadamard manifold with strictly negative sectional curvatures. Ark. Mat. 52 (2014), no. 1, 61--92. doi:10.1007/s11512-013-0182-3. https://projecteuclid.org/euclid.afm/1485802656

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