Some rigidity results for complete spacelike hypersurfaces with constant mean curvature in certain Lorentzian spaces Alma L. Albujer (joint work with Fernanda E. C. Camargo and Henrique F. de Lima) University of C´ ordoba Conference on Pure and Applied Differential Geometry, PADGE 2012 Leuven, August 27 ∼ 30, 2012
The results contained in this talk are partially contained in ⋆ A. L. Albujer, F. E. C. Camargo and H. F. de Lima, Complete spacelike hypersurfaces with constant mean curvature in H n × R 1 , J. Math. Anal. Appl. 368 (2010), 650–657. ⋆ A. L. Albujer, F. E. C. Camargo and H. F. de Lima, Complete spacelike hypersurfaces in a Robertson-Walker spacetime, Math. Proc. Camb. Phil. Soc. 151 (2011), 271–282. Alma L. Albujer (with F. E. C. Camargo and H. F. de Lima) Spacelike CMC hypersurfaces
Introduction In 1970 Calabi proved the well-known Calabi-Bernstein theorem: the only entire maximal graphs in R 3 1 are the spacelike planes. Alma L. Albujer (with F. E. C. Camargo and H. F. de Lima) Spacelike CMC hypersurfaces
Introduction In 1970 Calabi proved the well-known Calabi-Bernstein theorem: the only entire maximal graphs in R 3 1 are the spacelike planes. Equivalently, the only complete maximal surfaces in R 3 1 are the spacelike planes. Alma L. Albujer (with F. E. C. Camargo and H. F. de Lima) Spacelike CMC hypersurfaces
Introduction In 1970 Calabi proved the well-known Calabi-Bernstein theorem: the only entire maximal graphs in R 3 1 are the spacelike planes. Equivalently, the only complete maximal surfaces in R 3 1 are the spacelike planes. From then on many authors have studied existence, uniqueness and/or non-existence results for maximal and, in general, for spacelike constant mean curvature hypersurfaces in certain Lorentzian spaces. Alma L. Albujer (with F. E. C. Camargo and H. F. de Lima) Spacelike CMC hypersurfaces
Introduction In 1970 Calabi proved the well-known Calabi-Bernstein theorem: the only entire maximal graphs in R 3 1 are the spacelike planes. Equivalently, the only complete maximal surfaces in R 3 1 are the spacelike planes. From then on many authors have studied existence, uniqueness and/or non-existence results for maximal and, in general, for spacelike constant mean curvature hypersurfaces in certain Lorentzian spaces. In a joint work with Al´ ıas (Albujer-Al´ ıas, 2009), we have obtained Calabi-Bernstein type results for maximal surfaces in M 2 × R 1 under the assumption K M ≥ 0. Our results are no longer true in H 2 × R 1 (Albujer-Al´ ıas, 2009 and Albujer, 2008). Alma L. Albujer (with F. E. C. Camargo and H. F. de Lima) Spacelike CMC hypersurfaces
Introduction In 1970 Calabi proved the well-known Calabi-Bernstein theorem: the only entire maximal graphs in R 3 1 are the spacelike planes. Equivalently, the only complete maximal surfaces in R 3 1 are the spacelike planes. From then on many authors have studied existence, uniqueness and/or non-existence results for maximal and, in general, for spacelike constant mean curvature hypersurfaces in certain Lorentzian spaces. In a joint work with Al´ ıas (Albujer-Al´ ıas, 2009), we have obtained Calabi-Bernstein type results for maximal surfaces in M 2 × R 1 under the assumption K M ≥ 0. Our results are no longer true in H 2 × R 1 (Albujer-Al´ ıas, 2009 and Albujer, 2008). Recently, Alarc´ on and Souam have obtained examples of non-trivial entire spacelike graphs with constant mean curvature in H 2 × R 1 . Alma L. Albujer (with F. E. C. Camargo and H. F. de Lima) Spacelike CMC hypersurfaces
Introduction In 1970 Calabi proved the well-known Calabi-Bernstein theorem: the only entire maximal graphs in R 3 1 are the spacelike planes. Equivalently, the only complete maximal surfaces in R 3 1 are the spacelike planes. From then on many authors have studied existence, uniqueness and/or non-existence results for maximal and, in general, for spacelike constant mean curvature hypersurfaces in certain Lorentzian spaces. In a joint work with Al´ ıas (Albujer-Al´ ıas, 2009), we have obtained Calabi-Bernstein type results for maximal surfaces in M 2 × R 1 under the assumption K M ≥ 0. Our results are no longer true in H 2 × R 1 (Albujer-Al´ ıas, 2009 and Albujer, 2008). Recently, Alarc´ on and Souam have obtained examples of non-trivial entire spacelike graphs with constant mean curvature in H 2 × R 1 . Therefore, in order to obtain rigidity results for spacelike constant mean curvature surfaces of H 2 × R 1 , or in general hypersurfaces of H n × R 1 , we will need to ask some extra assumptions. Alma L. Albujer (with F. E. C. Camargo and H. F. de Lima) Spacelike CMC hypersurfaces
The Omori-Yau maximum principle Let f : [ a , b ] → R be a continuous function. Then f attains its maximum at some point p 0 ∈ [ a , b ]. Alma L. Albujer (with F. E. C. Camargo and H. F. de Lima) Spacelike CMC hypersurfaces
The Omori-Yau maximum principle Let f : [ a , b ] → R be a continuous function. Then f attains its maximum at some point p 0 ∈ [ a , b ]. ⋆ If p 0 ∈ ( a , b ) and f has continuous second derivative in a neighbourhood of p 0 , then f ′ ( p 0 ) = 0 f ′′ ( p 0 ) ≤ 0 . and Alma L. Albujer (with F. E. C. Camargo and H. F. de Lima) Spacelike CMC hypersurfaces
The Omori-Yau maximum principle Let f : [ a , b ] → R be a continuous function. Then f attains its maximum at some point p 0 ∈ [ a , b ]. ⋆ If p 0 ∈ ( a , b ) and f has continuous second derivative in a neighbourhood of p 0 , then f ′ ( p 0 ) = 0 f ′′ ( p 0 ) ≤ 0 . and Consider now a compact Riemannian manifold M (without boundary) and consider any smooth function f ∈ C 2 ( M ). Then f attains its maximum at some point p 0 ∈ M and |∇ f ( p 0 ) | = 0 and ∆ f ( p 0 ) ≤ 0 . Alma L. Albujer (with F. E. C. Camargo and H. F. de Lima) Spacelike CMC hypersurfaces
The Omori-Yau maximum principle Let f : [ a , b ] → R be a continuous function. Then f attains its maximum at some point p 0 ∈ [ a , b ]. ⋆ If p 0 ∈ ( a , b ) and f has continuous second derivative in a neighbourhood of p 0 , then f ′ ( p 0 ) = 0 f ′′ ( p 0 ) ≤ 0 . and Consider now a compact Riemannian manifold M (without boundary) and consider any smooth function f ∈ C 2 ( M ). Then f attains its maximum at some point p 0 ∈ M and |∇ f ( p 0 ) | = 0 and ∆ f ( p 0 ) ≤ 0 . When M is not compact, a given function f ∈ C 2 ( M ) with sup M f < + ∞ does not necessarily attains its supremum. Alma L. Albujer (with F. E. C. Camargo and H. F. de Lima) Spacelike CMC hypersurfaces
The Omori-Yau maximum principle Omori-Yau maximum principle (Omori, 1967 and Yau, 1975) Let M n be an n -dimensional complete Riemannian manifold whose Ricci curvature is bounded from below and consider f : M n → R a smooth function which is bounded from above on M n . Then there is a sequence of points { p k } k ∈ N ⊂ M n such that k →∞ f ( p k ) = sup lim f , k →∞ |∇ f ( p k ) | = 0 lim and k →∞ ∆ f ( p k ) ≤ 0 . lim M Alma L. Albujer (with F. E. C. Camargo and H. F. de Lima) Spacelike CMC hypersurfaces
Preliminaries for spacelike hypersurfaces in H n × R 1 Let Σ n be a spacelike hypersurface immersed into the Lorentzian product H n × R 1 . Alma L. Albujer (with F. E. C. Camargo and H. F. de Lima) Spacelike CMC hypersurfaces
Preliminaries for spacelike hypersurfaces in H n × R 1 Let Σ n be a spacelike hypersurface immersed into the Lorentzian product H n × R 1 . ⋆ Since ∂ t = ( ∂/∂ t ) ( x , t ) is a unitary timelike vector field globally defined on H n × R 1 , then there exists a unique timelike unitary normal vector field N globally defined on Σ n such that Σ n . � N , ∂ t � ≤ − 1 < 0 on Alma L. Albujer (with F. E. C. Camargo and H. F. de Lima) Spacelike CMC hypersurfaces
Preliminaries for spacelike hypersurfaces in H n × R 1 Let Σ n be a spacelike hypersurface immersed into the Lorentzian product H n × R 1 . ⋆ Since ∂ t = ( ∂/∂ t ) ( x , t ) is a unitary timelike vector field globally defined on H n × R 1 , then there exists a unique timelike unitary normal vector field N globally defined on Σ n such that Σ n . � N , ∂ t � ≤ − 1 < 0 on N is called the future-pointing Gauss map of Σ n and cosh θ = −� N , ∂ t � measures the normal hyperbolic angle of Σ n . Alma L. Albujer (with F. E. C. Camargo and H. F. de Lima) Spacelike CMC hypersurfaces
Preliminaries for spacelike hypersurfaces in H n × R 1 Let Σ n be a spacelike hypersurface immersed into the Lorentzian product H n × R 1 . ⋆ Since ∂ t = ( ∂/∂ t ) ( x , t ) is a unitary timelike vector field globally defined on H n × R 1 , then there exists a unique timelike unitary normal vector field N globally defined on Σ n such that Σ n . � N , ∂ t � ≤ − 1 < 0 on N is called the future-pointing Gauss map of Σ n and cosh θ = −� N , ∂ t � measures the normal hyperbolic angle of Σ n . Let h denote the height function of Σ n , h = ( π R ) ⊤ . It is not difficult to see that �∇ h � 2 = cosh 2 θ − 1 . ∇ h = − ∂ ⊤ t = − ∂ t + cosh θ N and Alma L. Albujer (with F. E. C. Camargo and H. F. de Lima) Spacelike CMC hypersurfaces
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