A Bernstein-type theorem for Riemannian manifolds with a Killing field

  • Luis J. Alías
  • Marcos Dajczer
  • Jaime Ripoll
Original Paper


The classical Bernstein theorem asserts that any complete minimal surface in Euclidean space \(\mathbb{R}^3\) that can be written as the graph of a function on \(\mathbb{R}^2\) must be a plane. In this paper, we extend Bernstein’s result to complete minimal surfaces in (may be non-complete) ambient spaces of non-negative Ricci curvature carrying a Killing field. This is done under the assumption that the sign of the angle function between a global Gauss map and the Killing field remains unchanged along the surface. In fact, our main result only requires the presence of a homothetic Killing field.


Complete minimal surface Bernstein theorem Homothetic Killing field 

Mathematics Subject Classifications

Primary 53A10 Secondary 53C42 


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Copyright information

© Springer Science + Business Media B.V. 2006

Authors and Affiliations

  1. 1.Departamento de MatemáticasUniversidad de MurciaEspinardoSpain
  2. 2.IMPARio de JaneiroBrazil
  3. 3.Instituto de MatemáticaUniversidade Federal do Rio Grande do SulPorto AlegreBrazil

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