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The Quarterly Journal of Mathematics Advance Access originally published online on June 26, 2007
The Quarterly Journal of Mathematics 2007 58(4):479-487; doi:10.1093/qmath/ham020
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© 2007. Published by Oxford University Press. All rights reserved. For permissions, please email: journals.permissions@oxfordjournals.org

AN ELEMENTARY PROOF OF THE ABRESCH–ROSENBERG THEOREM ON CONSTANT MEAN CURVATURE IMMERSED SURFACES IN S2 x R AND H2 x R

Maria Luiza Leite{dagger}

Departamento de Matemática, UFPe, Recife, 50.740-540 PE, Brasil

{dagger} E-mail: mll{at}dmat.ufpe.br

Received 25 August 2006; revised 1 March 2007
  Abstract

We make explicit the centers and radii of the horizontal geodesic circles on a constant mean curvature surface with null Abresch–Rosenberg differential in S2 x R and in H2 x R (horizontal horocycles are also determined) and prove that those centers project on to the same point, unless the complete surface is foliated by horocycles. This new visualization of the rotational and special surfaces classified by Abresch and Rosenberg is obtained in a direct way, just taking covariant derivatives of the unit normal along the flows of two global tangent fields. Moreover, this approach reveals that the special surfaces in H2 x R have constant intrinsic curvature K {equiv} –1+4H2 isin (–1, 0], so they form a non-rotational family of hyperbolic examples converging to a flat one, as 4H2 {nearrow} 1.


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