Some rigidity results for complete spacelike hypersurfaces with - - PowerPoint PPT Presentation
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
Alma L. Albujer (joint work with Fernanda E. C. Camargo and Henrique F. de Lima)
University of C´
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 Hn × R1, 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,
Alma L. Albujer (with F. E. C. Camargo and H. F. de Lima) Spacelike CMC hypersurfaces
In 1970 Calabi proved the well-known Calabi-Bernstein theorem: the
1 are the spacelike planes.
Alma L. Albujer (with F. E. C. Camargo and H. F. de Lima) Spacelike CMC hypersurfaces
In 1970 Calabi proved the well-known Calabi-Bernstein theorem: the
1 are the spacelike planes.
Equivalently, the only complete maximal surfaces in R3
1 are the
spacelike planes.
Alma L. Albujer (with F. E. C. Camargo and H. F. de Lima) Spacelike CMC hypersurfaces
In 1970 Calabi proved the well-known Calabi-Bernstein theorem: the
1 are the spacelike planes.
Equivalently, the only complete maximal surfaces in R3
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
In 1970 Calabi proved the well-known Calabi-Bernstein theorem: the
1 are the spacelike planes.
Equivalently, the only complete maximal surfaces in R3
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 M2 × R1 under the assumption KM ≥ 0. Our results are no longer true in H2 × R1 (Albujer-Al´ ıas, 2009 and Albujer, 2008).
Alma L. Albujer (with F. E. C. Camargo and H. F. de Lima) Spacelike CMC hypersurfaces
In 1970 Calabi proved the well-known Calabi-Bernstein theorem: the
1 are the spacelike planes.
Equivalently, the only complete maximal surfaces in R3
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 M2 × R1 under the assumption KM ≥ 0. Our results are no longer true in H2 × R1 (Albujer-Al´ ıas, 2009 and Albujer, 2008). Recently, Alarc´
entire spacelike graphs with constant mean curvature in H2 × R1.
Alma L. Albujer (with F. E. C. Camargo and H. F. de Lima) Spacelike CMC hypersurfaces
In 1970 Calabi proved the well-known Calabi-Bernstein theorem: the
1 are the spacelike planes.
Equivalently, the only complete maximal surfaces in R3
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 M2 × R1 under the assumption KM ≥ 0. Our results are no longer true in H2 × R1 (Albujer-Al´ ıas, 2009 and Albujer, 2008). Recently, Alarc´
entire spacelike graphs with constant mean curvature in H2 × R1. Therefore, in order to obtain rigidity results for spacelike constant mean curvature surfaces of H2 × R1, or in general hypersurfaces of Hn × R1, we will need to ask some extra assumptions.
Alma L. Albujer (with F. E. C. Camargo and H. F. de Lima) Spacelike CMC hypersurfaces
Let f : [a, b] → R be a continuous function. Then f attains its maximum at some point p0 ∈ [a, b].
Alma L. Albujer (with F. E. C. Camargo and H. F. de Lima) Spacelike CMC hypersurfaces
Let f : [a, b] → R be a continuous function. Then f attains its maximum at some point p0 ∈ [a, b]. ⋆ If p0 ∈ (a, b) and f has continuous second derivative in a neighbourhood of p0, then f ′(p0) = 0 and f ′′(p0) ≤ 0.
Alma L. Albujer (with F. E. C. Camargo and H. F. de Lima) Spacelike CMC hypersurfaces
Let f : [a, b] → R be a continuous function. Then f attains its maximum at some point p0 ∈ [a, b]. ⋆ If p0 ∈ (a, b) and f has continuous second derivative in a neighbourhood of p0, then f ′(p0) = 0 and f ′′(p0) ≤ 0. Consider now a compact Riemannian manifold M (without boundary) and consider any smooth function f ∈ C2(M). Then f attains its maximum at some point p0 ∈ M and |∇f (p0)| = 0 and ∆f (p0) ≤ 0.
Alma L. Albujer (with F. E. C. Camargo and H. F. de Lima) Spacelike CMC hypersurfaces
Let f : [a, b] → R be a continuous function. Then f attains its maximum at some point p0 ∈ [a, b]. ⋆ If p0 ∈ (a, b) and f has continuous second derivative in a neighbourhood of p0, then f ′(p0) = 0 and f ′′(p0) ≤ 0. Consider now a compact Riemannian manifold M (without boundary) and consider any smooth function f ∈ C2(M). Then f attains its maximum at some point p0 ∈ M and |∇f (p0)| = 0 and ∆f (p0) ≤ 0. When M is not compact, a given function f ∈ C2(M) with supM f < +∞ does not necessarily attains its supremum.
Alma L. Albujer (with F. E. C. Camargo and H. F. de Lima) Spacelike CMC hypersurfaces
Omori-Yau maximum principle (Omori, 1967 and Yau, 1975) Let Mn be an n-dimensional complete Riemannian manifold whose Ricci curvature is bounded from below and consider f : Mn → R a smooth function which is bounded from above on Mn. Then there is a sequence
lim
k→∞ f (pk) = sup M
f , lim
k→∞ |∇f (pk)| = 0
and lim
k→∞ ∆f (pk) ≤ 0.
Alma L. Albujer (with F. E. C. Camargo and H. F. de Lima) Spacelike CMC hypersurfaces
Let Σn be a spacelike hypersurface immersed into the Lorentzian product Hn × R1.
Alma L. Albujer (with F. E. C. Camargo and H. F. de Lima) Spacelike CMC hypersurfaces
Let Σn be a spacelike hypersurface immersed into the Lorentzian product Hn × R1. ⋆ Since ∂t = (∂/∂t)(x,t) is a unitary timelike vector field globally defined on Hn × R1, then there exists a unique timelike unitary normal vector field N globally defined on Σn such that N, ∂t ≤ −1 < 0
Σn.
Alma L. Albujer (with F. E. C. Camargo and H. F. de Lima) Spacelike CMC hypersurfaces
Let Σn be a spacelike hypersurface immersed into the Lorentzian product Hn × R1. ⋆ Since ∂t = (∂/∂t)(x,t) is a unitary timelike vector field globally defined on Hn × R1, then there exists a unique timelike unitary normal vector field N globally defined on Σn such that N, ∂t ≤ −1 < 0
Σn. N is called the future-pointing Gauss map of Σn and cosh θ = −N, ∂t measures the normal hyperbolic angle of Σn.
Alma L. Albujer (with F. E. C. Camargo and H. F. de Lima) Spacelike CMC hypersurfaces
Let Σn be a spacelike hypersurface immersed into the Lorentzian product Hn × R1. ⋆ Since ∂t = (∂/∂t)(x,t) is a unitary timelike vector field globally defined on Hn × R1, then there exists a unique timelike unitary normal vector field N globally defined on Σn such that N, ∂t ≤ −1 < 0
Σn. N is called the future-pointing Gauss map of Σn and cosh θ = −N, ∂t measures the normal hyperbolic angle of Σn. Let h denote the height function of Σn, h = (πR)⊤. It is not difficult to see that ∇h = −∂⊤
t = −∂t + cosh θ N
and ∇h2 = cosh2 θ − 1.
Alma L. Albujer (with F. E. C. Camargo and H. F. de Lima) Spacelike CMC hypersurfaces
Let Σn be a spacelike hypersurface immersed into the Lorentzian product Hn × R1. ⋆ Since ∂t = (∂/∂t)(x,t) is a unitary timelike vector field globally defined on Hn × R1, then there exists a unique timelike unitary normal vector field N globally defined on Σn such that N, ∂t ≤ −1 < 0
Σn. N is called the future-pointing Gauss map of Σn and cosh θ = −N, ∂t measures the normal hyperbolic angle of Σn. Let h denote the height function of Σn, h = (πR)⊤. It is not difficult to see that ∇h = −∂⊤
t = −∂t + cosh θ N
and ∇h2 = cosh2 θ − 1. Given t0 ∈ R, the spacelike hypersurface Hn × {t0} is called a slice. Slices are characterized as the spacelike hypersurfaces with θ ≡ 0. Or equivalently, as the spacelike hypersurfaces with constant height function.
Alma L. Albujer (with F. E. C. Camargo and H. F. de Lima) Spacelike CMC hypersurfaces
Let A : X(Σ) → X(Σ) be the shape operator of Σn with respect to N and let λ1, ..., λn be the principal curvatures of Σn.
Alma L. Albujer (with F. E. C. Camargo and H. F. de Lima) Spacelike CMC hypersurfaces
Let A : X(Σ) → X(Σ) be the shape operator of Σn with respect to N and let λ1, ..., λn be the principal curvatures of Σn. The r-mean curvature function Hr of the spacelike hypersurface Σn is defined by n r
λi1(p) · · · λir (p) , 1 ≤ r ≤ n
Alma L. Albujer (with F. E. C. Camargo and H. F. de Lima) Spacelike CMC hypersurfaces
Let A : X(Σ) → X(Σ) be the shape operator of Σn with respect to N and let λ1, ..., λn be the principal curvatures of Σn. The r-mean curvature function Hr of the spacelike hypersurface Σn is defined by n r
λi1(p) · · · λir (p) , 1 ≤ r ≤ n ⋆ In the particular case when r = 1, H1 = H = − 1
ntr(A) is the mean
curvature of Σn. ⋆ In the case r = 2, H2 defines a geometric quantity related to the scalar curvature of the hypersurface.
Alma L. Albujer (with F. E. C. Camargo and H. F. de Lima) Spacelike CMC hypersurfaces
Theorem Let Σn be a complete spacelike hypersurface of Hn × R1 with constant mean curvature H. If the height function h of Σn satisfies ∇h2 ≤ nα n − 1H2, for some constant 0 < α < 1, then Σn is a slice.
Alma L. Albujer (with F. E. C. Camargo and H. F. de Lima) Spacelike CMC hypersurfaces
Theorem Let Σn be a complete spacelike hypersurface of Hn × R1 with constant mean curvature H. If the height function h of Σn satisfies ∇h2 ≤ nα n − 1H2, for some constant 0 < α < 1, then Σn is a slice. Or equivalently, Corollary There do not exist any complete spacelike hypersurface in Hn × R1 with constant mean curvature H = 0 such that ∇h2 ≤ nα n − 1H2. for any constant 0 < α < 1.
Alma L. Albujer (with F. E. C. Camargo and H. F. de Lima) Spacelike CMC hypersurfaces
Sketch of the proof: We will apply the Omori-Yau maximum principle to cosh θ.
Alma L. Albujer (with F. E. C. Camargo and H. F. de Lima) Spacelike CMC hypersurfaces
Sketch of the proof: We will apply the Omori-Yau maximum principle to cosh θ. ⋆ cosh θ is bounded from above since ∇h2 ≤
nα n−1H2.
Alma L. Albujer (with F. E. C. Camargo and H. F. de Lima) Spacelike CMC hypersurfaces
Sketch of the proof: We will apply the Omori-Yau maximum principle to cosh θ. ⋆ cosh θ is bounded from above since ∇h2 ≤
nα n−1H2.
⋆ By the Gauss equation, for any X ∈ X(Σ) it holds that Ric(X, X) = − (n − 1)X2 − (n − 2)X, ∇h2 − ∇h2X2 +
2 X
− n2H2 4 X2
Alma L. Albujer (with F. E. C. Camargo and H. F. de Lima) Spacelike CMC hypersurfaces
Sketch of the proof: We will apply the Omori-Yau maximum principle to cosh θ. ⋆ cosh θ is bounded from above since ∇h2 ≤
nα n−1H2.
⋆ By the Gauss equation, for any X ∈ X(Σ) it holds that Ric(X, X) = − (n − 1)X2 − (n − 2)X, ∇h2 − ∇h2X2 +
2 X
− n2H2 4 X2 ≥−
4 + (n − 1)∇h2
Alma L. Albujer (with F. E. C. Camargo and H. F. de Lima) Spacelike CMC hypersurfaces
Sketch of the proof: We will apply the Omori-Yau maximum principle to cosh θ. ⋆ cosh θ is bounded from above since ∇h2 ≤
nα n−1H2.
⋆ By the Gauss equation, for any X ∈ X(Σ) it holds that Ric(X, X) = − (n − 1)X2 − (n − 2)X, ∇h2 − ∇h2X2 +
2 X
− n2H2 4 X2 ≥−
4 + (n − 1)∇h2
is bounded from above.
Alma L. Albujer (with F. E. C. Camargo and H. F. de Lima) Spacelike CMC hypersurfaces
Sketch of the proof: We will apply the Omori-Yau maximum principle to cosh θ. ⋆ cosh θ is bounded from above since ∇h2 ≤
nα n−1H2.
⋆ By the Gauss equation, for any X ∈ X(Σ) it holds that Ric(X, X) = − (n − 1)X2 − (n − 2)X, ∇h2 − ∇h2X2 +
2 X
− n2H2 4 X2 ≥−
4 + (n − 1)∇h2
is bounded from above. There exists a sequence {pk}k∈N on Σn such that lim
k→∞ cosh θ(pk) = sup p∈Σn cosh θ(p)
and lim
k→∞ ∆ cosh θ(pk) ≤ 0.
Alma L. Albujer (with F. E. C. Camargo and H. F. de Lima) Spacelike CMC hypersurfaces
Sketch of the proof: On the other hand, we can compute ∆ cosh θ =
cosh θ.
Alma L. Albujer (with F. E. C. Camargo and H. F. de Lima) Spacelike CMC hypersurfaces
Sketch of the proof: On the other hand, we can compute ∆ cosh θ =
cosh θ. ⋆ By the Cauchy-Schwarz inequality H2 − H2 ≥ 0. Therefore, ∆ cosh θ =
cosh θ ≥n(1 − α)H2 cosh θ ≥ 0.
Alma L. Albujer (with F. E. C. Camargo and H. F. de Lima) Spacelike CMC hypersurfaces
Sketch of the proof: On the other hand, we can compute ∆ cosh θ =
cosh θ. ⋆ By the Cauchy-Schwarz inequality H2 − H2 ≥ 0. Therefore, ∆ cosh θ =
cosh θ ≥n(1 − α)H2 cosh θ ≥ 0. As a consequence we have the chain of inequalities 0 ≤ n(1 − α)H2 sup
p∈Σn cosh θ(p) ≤ lim k→∞ ∆ cosh θ(pk)
Alma L. Albujer (with F. E. C. Camargo and H. F. de Lima) Spacelike CMC hypersurfaces
Sketch of the proof: On the other hand, we can compute ∆ cosh θ =
cosh θ. ⋆ By the Cauchy-Schwarz inequality H2 − H2 ≥ 0. Therefore, ∆ cosh θ =
cosh θ ≥n(1 − α)H2 cosh θ ≥ 0. As a consequence we have the chain of inequalities 0 ≤ n(1 − α)H2 sup
p∈Σn cosh θ(p) ≤ lim k→∞ ∆ cosh θ(pk) ≤ 0
Alma L. Albujer (with F. E. C. Camargo and H. F. de Lima) Spacelike CMC hypersurfaces
Sketch of the proof: On the other hand, we can compute ∆ cosh θ =
cosh θ. ⋆ By the Cauchy-Schwarz inequality H2 − H2 ≥ 0. Therefore, ∆ cosh θ =
cosh θ ≥n(1 − α)H2 cosh θ ≥ 0. As a consequence we have the chain of inequalities 0 ≤ n(1 − α)H2 sup
p∈Σn cosh θ(p) ≤ lim k→∞ ∆ cosh θ(pk) ≤ 0
Alma L. Albujer (with F. E. C. Camargo and H. F. de Lima) Spacelike CMC hypersurfaces
Theorem Let Σn be a complete spacelike hypersurface of Hn × R1 with constant mean curvature H and H2 bounded from below. If the height function h
∇h2 ≤ α n − 1A2, for some constant 0 < α < 1, where A2 denotes the squared norm of the shape operator A, then Σn is a slice.
Alma L. Albujer (with F. E. C. Camargo and H. F. de Lima) Spacelike CMC hypersurfaces
Theorem Let Σn be a complete spacelike hypersurface of Hn × R1 with constant mean curvature H and H2 bounded from below. If the height function h
∇h2 ≤ α n − 1A2, for some constant 0 < α < 1, where A2 denotes the squared norm of the shape operator A, then Σn is a slice. ⋆ Observe that nH2 ≤ A2 and, consequently, the assumption ∇h2 ≤
α n−1A2 is less restrictive than ∇h2 ≤ nα n−1H2.
Alma L. Albujer (with F. E. C. Camargo and H. F. de Lima) Spacelike CMC hypersurfaces
Theorem Let Σn be a complete spacelike hypersurface of Hn × R1 with constant mean curvature H and H2 bounded from below. If the height function h
∇h2 ≤ α n − 1A2, for some constant 0 < α < 1, where A2 denotes the squared norm of the shape operator A, then Σn is a slice. ⋆ Observe that nH2 ≤ A2 and, consequently, the assumption ∇h2 ≤
α n−1A2 is less restrictive than ∇h2 ≤ nα n−1H2.
⋆ The proof of this theorem is analogous to the previous one. However, now A2 is not necessarily constant, but since it holds A2 = n2H2 − n(n − 1)H2 we can guarantee that supΣ A2 < +∞. This is enough for our purpose.
Alma L. Albujer (with F. E. C. Camargo and H. F. de Lima) Spacelike CMC hypersurfaces
An entire graph in Hn × R1 is determined by a smooth function u ∈ C∞(Hn) and it is given by Σn(u) = {(x, u(x)) : x ∈ Hn} ⊂ Hn × R1.
Alma L. Albujer (with F. E. C. Camargo and H. F. de Lima) Spacelike CMC hypersurfaces
An entire graph in Hn × R1 is determined by a smooth function u ∈ C∞(Hn) and it is given by Σn(u) = {(x, u(x)) : x ∈ Hn} ⊂ Hn × R1. ⋆ The metric induced on Hn from the Lorentzian metric of Hn × R1 via the graph is , = −du2 + , Hn.
Alma L. Albujer (with F. E. C. Camargo and H. F. de Lima) Spacelike CMC hypersurfaces
An entire graph in Hn × R1 is determined by a smooth function u ∈ C∞(Hn) and it is given by Σn(u) = {(x, u(x)) : x ∈ Hn} ⊂ Hn × R1. ⋆ The metric induced on Hn from the Lorentzian metric of Hn × R1 via the graph is , = −du2 + , Hn. ⋆ The graph Σn(u) is a spacelike hypersurface if and only if |Du|2 < 1.
Alma L. Albujer (with F. E. C. Camargo and H. F. de Lima) Spacelike CMC hypersurfaces
An entire graph in Hn × R1 is determined by a smooth function u ∈ C∞(Hn) and it is given by Σn(u) = {(x, u(x)) : x ∈ Hn} ⊂ Hn × R1. ⋆ The metric induced on Hn from the Lorentzian metric of Hn × R1 via the graph is , = −du2 + , Hn. ⋆ The graph Σn(u) is a spacelike hypersurface if and only if |Du|2 < 1. For any entire spacelike graph we have the following relation: ∇h2 = |Du|2 1 − |Du|2 .
Alma L. Albujer (with F. E. C. Camargo and H. F. de Lima) Spacelike CMC hypersurfaces
An entire graph in Hn × R1 is determined by a smooth function u ∈ C∞(Hn) and it is given by Σn(u) = {(x, u(x)) : x ∈ Hn} ⊂ Hn × R1. ⋆ The metric induced on Hn from the Lorentzian metric of Hn × R1 via the graph is , = −du2 + , Hn. ⋆ The graph Σn(u) is a spacelike hypersurface if and only if |Du|2 < 1. For any entire spacelike graph we have the following relation: ∇h2 = |Du|2 1 − |Du|2 . It is well known that an entire spacelike graph is not necessarily
constant c such that |Du|2 < c < 1, then the graph is complete.
Alma L. Albujer (with F. E. C. Camargo and H. F. de Lima) Spacelike CMC hypersurfaces
Theorem Let Σn(u) be an entire spacelike graph in Hn × R1 with constant mean curvature H. If the function u satisfies |Du|2 ≤ nαH2 n − 1 + nαH2 for some constant 0 < α < 1, then Σn(u) is a slice.
Alma L. Albujer (with F. E. C. Camargo and H. F. de Lima) Spacelike CMC hypersurfaces
Theorem Let Σn(u) be an entire spacelike graph in Hn × R1 with constant mean curvature H. If the function u satisfies |Du|2 ≤ nαH2 n − 1 + nαH2 for some constant 0 < α < 1, then Σn(u) is a slice. Corollary There do not exist any entire spacelike graph in Hn × R1 with constant mean curvature H = 0 such that |Du|2 ≤ nαH2 n − 1 + nαH2 for any constant 0 < α < 1.
Alma L. Albujer (with F. E. C. Camargo and H. F. de Lima) Spacelike CMC hypersurfaces
Theorem Let Σn(u) be an entire spacelike graph in Hn × R1 with constant mean curvature H and H2 bounded from below. If the function u is such that |Du|2 ≤ αA2 n − 1 + αA2 , for some constant 0 < α < 1, then Σn(u) is a slice.
Alma L. Albujer (with F. E. C. Camargo and H. F. de Lima) Spacelike CMC hypersurfaces
Let (Mn, , M) be a Riemannian manifold of constant sectional curvature κ, I ⊆ R a real interval and f > 0 a positive smooth function on I.
Alma L. Albujer (with F. E. C. Camargo and H. F. de Lima) Spacelike CMC hypersurfaces
Let (Mn, , M) be a Riemannian manifold of constant sectional curvature κ, I ⊆ R a real interval and f > 0 a positive smooth function on I. A Robertson-Walker spacetime is the product manifold Mn × I endowed with the Lorentzian metric , = f 2, M − dt2. We denote it by Mn
f × I1.
Alma L. Albujer (with F. E. C. Camargo and H. F. de Lima) Spacelike CMC hypersurfaces
Let (Mn, , M) be a Riemannian manifold of constant sectional curvature κ, I ⊆ R a real interval and f > 0 a positive smooth function on I. A Robertson-Walker spacetime is the product manifold Mn × I endowed with the Lorentzian metric , = f 2, M − dt2. We denote it by Mn
f × I1.
Some energy conditions A spacetime M
n+1 satisfies the timelike convergence condition
(TCC) if Ric(Z, Z) ≥ 0, for all timelike vector Z.
Alma L. Albujer (with F. E. C. Camargo and H. F. de Lima) Spacelike CMC hypersurfaces
Let (Mn, , M) be a Riemannian manifold of constant sectional curvature κ, I ⊆ R a real interval and f > 0 a positive smooth function on I. A Robertson-Walker spacetime is the product manifold Mn × I endowed with the Lorentzian metric , = f 2, M − dt2. We denote it by Mn
f × I1.
Some energy conditions A spacetime M
n+1 satisfies the timelike convergence condition
(TCC) if Ric(Z, Z) ≥ 0, for all timelike vector Z. A spacetime M
n+1 satisfies the null convergence condition
(NCC) if Ric(Z, Z) ≥ 0, for all lightlike vector Z.
Alma L. Albujer (with F. E. C. Camargo and H. F. de Lima) Spacelike CMC hypersurfaces
Let (Mn, , M) be a Riemannian manifold of constant sectional curvature κ, I ⊆ R a real interval and f > 0 a positive smooth function on I. A Robertson-Walker spacetime is the product manifold Mn × I endowed with the Lorentzian metric , = f 2, M − dt2. We denote it by Mn
f × I1.
Some energy conditions ⋆ For the case of a Robertson-Walker spacetime: TCC ⇔ f ′′ ≤ 0 κ ≥ sup
I
(ff ′′ − f ′2)
Alma L. Albujer (with F. E. C. Camargo and H. F. de Lima) Spacelike CMC hypersurfaces
Let (Mn, , M) be a Riemannian manifold of constant sectional curvature κ, I ⊆ R a real interval and f > 0 a positive smooth function on I. A Robertson-Walker spacetime is the product manifold Mn × I endowed with the Lorentzian metric , = f 2, M − dt2. We denote it by Mn
f × I1.
Some energy conditions ⋆ For the case of a Robertson-Walker spacetime: TCC ⇔ f ′′ ≤ 0 κ ≥ sup
I
(ff ′′ − f ′2) NCC ⇔ κ ≥ sup
I
(ff ′′ − f ′2)
Alma L. Albujer (with F. E. C. Camargo and H. F. de Lima) Spacelike CMC hypersurfaces
As another application of the Omori-Yau maximum principle: Theorem Let Mn
f × I1 be a RW spacetime satisfying NCC. Let Σn be a complete
spacelike hypersurface having constant mean curvature H = 0 and contained in a slab Mn × [t1, t2]. If 0 ≤ H sup
Σ
f ′ f ◦ h
and ∇h2 ≤ α
Σ
f ′ f ◦ h
for some positive constants α and β, then Σn is a slice.
Alma L. Albujer (with F. E. C. Camargo and H. F. de Lima) Spacelike CMC hypersurfaces
As another application of the Omori-Yau maximum principle: Theorem Let Mn
f × I1 be a RW spacetime satisfying NCC. Let Σn be a complete
spacelike hypersurface having constant mean curvature H = 0 and contained in a slab Mn × [t1, t2]. If 0 ≤ H sup
Σ
f ′ f ◦ h
and ∇h2 ≤ α
Σ
f ′ f ◦ h
for some positive constants α and β, then Σn is a slice. ⋆ Key of the proof: Apply the Omori-Yau maximum principle to the function f (h) cosh θ.
Alma L. Albujer (with F. E. C. Camargo and H. F. de Lima) Spacelike CMC hypersurfaces
For the case of a static RW spacetime: Corollary Let Mn × I1 be a static RW spacetime whose Riemannian fiber Mn has non-negative constant sectional curvature. Let Σn be a complete spacelike hypersurface in Mn × I1 with constant mean curvature H. Suppose that the normal hyperbolic angle of Σn is bounded. Then, Σn is maximal.
Alma L. Albujer (with F. E. C. Camargo and H. F. de Lima) Spacelike CMC hypersurfaces
For the case of a static RW spacetime: Corollary Let Mn × I1 be a static RW spacetime whose Riemannian fiber Mn has non-negative constant sectional curvature. Let Σn be a complete spacelike hypersurface in Mn × I1 with constant mean curvature H. Suppose that the normal hyperbolic angle of Σn is bounded. Then, Σn is maximal. Theorem Let Mn × I1 be a static RW spacetime whose Riemmanian fiber Mn has non-negative constant sectional curvature. Let Σn be a complete spacelike hypersurface with constant mean curvature H and H2 bounded from below. If the height function h of Σn is such that ∇h2 ≤ αAβ, for some positive constants α and β, then Σn is a slice.
Alma L. Albujer (with F. E. C. Camargo and H. F. de Lima) Spacelike CMC hypersurfaces
Again, it is possible to give non-parametric versions of our results: Theorem Let Mn
f × I1 be a RW spacetime with complete fiber Mn, and suppose
that it obeys NCC. Let Σn(u) be an entire spacelike graph in Mn
f × I1
with constant mean curvature H = 0 and contained in a slab Mn × [t1, t2]. If 0 ≤ H sup
Σn(u)
f ′ f ◦ u
and |Du|2 ≤ α infΣn(u)(f 2 ◦ u)
f ◦ u)
1 + α
f ◦ u)
for some positive constants α and β, then Σn(u) is a slice.
Alma L. Albujer (with F. E. C. Camargo and H. F. de Lima) Spacelike CMC hypersurfaces
Theorem Let Mn × I1 be a static RW spacetime whose Riemannian fiber Mn is a complete manifold with non-negative constant sectional curvature. Let Σn(u) be an entire spacelike graph in Mn × I1 with constant mean curvature H, H2 bounded from below and bounded normal hyperbolic
|Du|2 ≤ α Aβ 1 + α Aβ , for some positive constants α and β, then Σn(u) is a slice.
Alma L. Albujer (with F. E. C. Camargo and H. F. de Lima) Spacelike CMC hypersurfaces
Theorem Let Mn × I1 be a static RW spacetime whose Riemannian fiber Mn is a complete manifold with non-negative constant sectional curvature. Let Σn(u) be an entire spacelike graph in Mn × I1 with constant mean curvature H, H2 bounded from below and bounded normal hyperbolic
|Du|2 ≤ α Aβ 1 + α Aβ , for some positive constants α and β, then Σn(u) is a slice.
Alma L. Albujer (with F. E. C. Camargo and H. F. de Lima) Spacelike CMC hypersurfaces