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''complete maximal space-like hypersurfaces in an''
 
''complete maximal space-like hypersurfaces in an''
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110620/a1106201.png" /> be an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110620/a1106202.png" />-dimensional [[Minkowski space|Minkowski space]] of index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110620/a1106203.png" />, i.e., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110620/a1106204.png" /> and is equipped with the Lorentz metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110620/a1106205.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110620/a1106206.png" />, let
+
Let $  \mathbf R _ {p + 1 }  ^ {n + p + 1 } $
 +
be an $  ( n + p + 1 ) $-
 +
dimensional [[Minkowski space|Minkowski space]] of index $  p + 1 $,  
 +
i.e., $  \mathbf R _ {p + 1 }  ^ {n + p + 1 } = \{ ( x _ {1} \dots x _ {n + p + 1 }  ) \in \mathbf R ^ {n + p + 1 } \} $
 +
and is equipped with the Lorentz metric $  \sum _ {i = 1 }  ^ {n} ( dx _ {i} )  ^ {2} - \sum _ {j = 1 }  ^ {p + 1 } ( dx _ {n + j }  )  ^ {2} $.  
 +
For  $  c > 0 $,  
 +
let
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110620/a1106207.png" /></td> </tr></table>
+
$$
 +
H _ {p} ^ {n + p } ( c ) = \{ {x \in \mathbf R _ {p + 1 }  ^ {n + p + 1 } }
 +
:  
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110620/a1106208.png" /></td> </tr></table>
+
$$
 +
\
 +
{} {x _ {1}  ^ {2} + \dots + x _ {n}  ^ {2} - x _ {n + 1 }  ^ {2} - \dots - x _ {n + p + 1 }  ^ {2} = - {1 / c }  } \} .
 +
$$
  
Thus, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110620/a1106209.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110620/a11062010.png" />-dimensional indefinite [[Riemannian manifold|Riemannian manifold]] of index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110620/a11062011.png" /> and of constant [[Curvature|curvature]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110620/a11062012.png" />. It is called an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110620/a11062013.png" />-dimensional anti-de Sitter space of constant curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110620/a11062014.png" /> and of index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110620/a11062015.png" />. A hypersurface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110620/a11062016.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110620/a11062017.png" /> is said to be space-like if the metric on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110620/a11062018.png" /> induced by that of ambient space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110620/a11062019.png" /> is positive definite. The mean curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110620/a11062020.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110620/a11062021.png" /> is defined as in the case of Riemannian manifolds. By definition, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110620/a11062022.png" /> is a maximal hypersurface if the mean curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110620/a11062023.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110620/a11062024.png" /> is identically zero. S. Ishihara proved that a complete maximal space-like hypersurface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110620/a11062025.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110620/a11062026.png" /> satisfies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110620/a11062027.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110620/a11062028.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110620/a11062029.png" /> is isometric to the hyperbolic cylinder <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110620/a11062030.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110620/a11062031.png" /> is the squared norm of the [[Second fundamental form|second fundamental form]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110620/a11062032.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110620/a11062033.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110620/a11062034.png" />, is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110620/a11062035.png" />-dimensional hyperbolic space of constant curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110620/a11062036.png" />. The rigidity of the hyperbolic cylinder <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110620/a11062037.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110620/a11062038.png" /> was proved by U.-H. Ki, H.S. Kim and H. Nakagawa [[#References|[a3]]]: for a given integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110620/a11062039.png" /> and constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110620/a11062040.png" />, there exists a constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110620/a11062041.png" />, depending on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110620/a11062042.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110620/a11062043.png" />, such that the hyperbolic cylinder <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110620/a11062044.png" /> is the only complete maximal space-like hypersurface in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110620/a11062045.png" /> of constant scalar curvature and such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110620/a11062046.png" />. In particular, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110620/a11062047.png" />, Q.M. Cheng [[#References|[a1]]] has characterized the complete maximal space-like hypersurfaces in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110620/a11062048.png" /> under the condition of constant Gauss–Kronecker curvature (cf. [[Gaussian curvature|Gaussian curvature]]): Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110620/a11062049.png" /> be a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110620/a11062050.png" />-dimensional complete maximal space-like hypersurface of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110620/a11062051.png" />. Now:
+
Thus, $  H _ {p} ^ {n + p } ( c ) $
 +
is an $  ( n + p ) $-
 +
dimensional indefinite [[Riemannian manifold|Riemannian manifold]] of index $  p $
 +
and of constant [[Curvature|curvature]] $  - c $.  
 +
It is called an $  ( n + p ) $-
 +
dimensional anti-de Sitter space of constant curvature $  - c $
 +
and of index $  p $.  
 +
A hypersurface $  M $
 +
of $  H _ {1} ^ {n + 1 } ( c ) $
 +
is said to be space-like if the metric on $  M $
 +
induced by that of ambient space $  H _ {1} ^ {n + 1 } ( c ) $
 +
is positive definite. The mean curvature $  H $
 +
of $  M $
 +
is defined as in the case of Riemannian manifolds. By definition, $  M $
 +
is a maximal hypersurface if the mean curvature $  H $
 +
of $  M $
 +
is identically zero. S. Ishihara proved that a complete maximal space-like hypersurface $  M $
 +
in $  H _ {1} ^ {n + 1 } ( c ) $
 +
satisfies $  S \leq  nc $,  
 +
and $  S = nc $
 +
if and only if $  M $
 +
is isometric to the hyperbolic cylinder $  H  ^ {k} ( c _ {1} ) \times H ^ {n - k } ( c _ {2} ) $,  
 +
where $  S $
 +
is the squared norm of the [[Second fundamental form|second fundamental form]] of $  M $
 +
and $  H  ^ {k} ( c _ {i} ) $,
 +
$  i = 1, 2 $,  
 +
is a $  k $-
 +
dimensional hyperbolic space of constant curvature $  c _ {i} $.  
 +
The rigidity of the hyperbolic cylinder $  H  ^ {k} ( c _ {1} ) \times H ^ {n - k } ( c _ {2} ) $
 +
in $  H _ {1} ^ {n + 1 } ( c ) $
 +
was proved by U.-H. Ki, H.S. Kim and H. Nakagawa [[#References|[a3]]]: for a given integer $  n $
 +
and constant $  c > 0 $,  
 +
there exists a constant $  C < nc $,  
 +
depending on $  n $
 +
and $  c $,  
 +
such that the hyperbolic cylinder $  H  ^ {k} ( c _ {1} ) \times H ^ {n - k } ( c _ {2} ) $
 +
is the only complete maximal space-like hypersurface in $  H _ {1} ^ {n + 1 } ( c ) $
 +
of constant scalar curvature and such that $  S > C $.  
 +
In particular, for $  n = 3 $,  
 +
Q.M. Cheng [[#References|[a1]]] has characterized the complete maximal space-like hypersurfaces in $  H _ {1}  ^ {4} ( c ) $
 +
under the condition of constant Gauss–Kronecker curvature (cf. [[Gaussian curvature|Gaussian curvature]]): Let $  M $
 +
be a $  3 $-
 +
dimensional complete maximal space-like hypersurface of $  H _ {1}  ^ {4} ( c ) $.  
 +
Now:
  
1) if the Gauss–Kronecker curvature of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110620/a11062052.png" /> is a non-zero constant, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110620/a11062053.png" /> is the hyperbolic cylinder <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110620/a11062054.png" />;
+
1) if the Gauss–Kronecker curvature of $  M $
 +
is a non-zero constant, then $  M $
 +
is the hyperbolic cylinder $  H  ^ {1} ( c _ {1} ) \times H  ^ {2} ( c _ {2} ) $;
  
2) if the [[Scalar curvature|scalar curvature]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110620/a11062055.png" /> is constant and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110620/a11062056.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110620/a11062057.png" /> is the hyperbolic cylinder <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110620/a11062058.png" />. There are no complete maximal space-like hypersurfaces in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110620/a11062059.png" /> with constant scalar curvature and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110620/a11062060.png" />.
+
2) if the [[Scalar curvature|scalar curvature]] $  K $
 +
is constant and $  \inf  K  ^ {2} > 0 $,  
 +
then $  M $
 +
is the hyperbolic cylinder $  H  ^ {1} ( c _ {1} ) \times H  ^ {2} ( c _ {2} ) $.  
 +
There are no complete maximal space-like hypersurfaces in $  H _ {1}  ^ {4} ( c ) $
 +
with constant scalar curvature and $  \sup  K  ^ {2} < { {S  ^ {3} } / {54 } } $.
  
 
On the other hand, complete space-like submanifolds in anti-de Sitter spaces with parallel mean curvature have been investigated by many authors.
 
On the other hand, complete space-like submanifolds in anti-de Sitter spaces with parallel mean curvature have been investigated by many authors.
Line 18: Line 92:
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  Q.M. Cheng,  "Complete maximal space-like hypersurfaces of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110620/a11062061.png" />"  ''Manuscr. Math.'' , '''82'''  (1994)  pp. 149–160</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  T. Ishikawa,  "Maximal space-like submanifolds of a pseudo–Riemannian space of constant curvature"  ''Michigan Math. J.'' , '''35'''  (1988)  pp. 345–352</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  U-H. Ki,  H.S. Kim,  H. Nakagawa,  "Complete maximal space-like hypersurfaces of an anti-de Sitter space"  ''Kyungpook Math. J.'' , '''31'''  (1991)  pp. 131–141</TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top">  Q.M. Cheng,  "Complete maximal space-like hypersurfaces of $H_1^4(c)$"  ''Manuscr. Math.'' , '''82'''  (1994)  pp. 149–160</TD></TR>
 +
<TR><TD valign="top">[a2]</TD> <TD valign="top">  T. Ishikawa,  "Maximal space-like submanifolds of a pseudo–Riemannian space of constant curvature"  ''Michigan Math. J.'' , '''35'''  (1988)  pp. 345–352</TD></TR>
 +
<TR><TD valign="top">[a3]</TD> <TD valign="top">  U-H. Ki,  H.S. Kim,  H. Nakagawa,  "Complete maximal space-like hypersurfaces of an anti-de Sitter space"  ''Kyungpook Math. J.'' , '''31'''  (1991)  pp. 131–141</TD></TR>
 +
</table>

Latest revision as of 06:52, 26 March 2023


complete maximal space-like hypersurfaces in an

Let $ \mathbf R _ {p + 1 } ^ {n + p + 1 } $ be an $ ( n + p + 1 ) $- dimensional Minkowski space of index $ p + 1 $, i.e., $ \mathbf R _ {p + 1 } ^ {n + p + 1 } = \{ ( x _ {1} \dots x _ {n + p + 1 } ) \in \mathbf R ^ {n + p + 1 } \} $ and is equipped with the Lorentz metric $ \sum _ {i = 1 } ^ {n} ( dx _ {i} ) ^ {2} - \sum _ {j = 1 } ^ {p + 1 } ( dx _ {n + j } ) ^ {2} $. For $ c > 0 $, let

$$ H _ {p} ^ {n + p } ( c ) = \{ {x \in \mathbf R _ {p + 1 } ^ {n + p + 1 } } : $$

$$ \ {} {x _ {1} ^ {2} + \dots + x _ {n} ^ {2} - x _ {n + 1 } ^ {2} - \dots - x _ {n + p + 1 } ^ {2} = - {1 / c } } \} . $$

Thus, $ H _ {p} ^ {n + p } ( c ) $ is an $ ( n + p ) $- dimensional indefinite Riemannian manifold of index $ p $ and of constant curvature $ - c $. It is called an $ ( n + p ) $- dimensional anti-de Sitter space of constant curvature $ - c $ and of index $ p $. A hypersurface $ M $ of $ H _ {1} ^ {n + 1 } ( c ) $ is said to be space-like if the metric on $ M $ induced by that of ambient space $ H _ {1} ^ {n + 1 } ( c ) $ is positive definite. The mean curvature $ H $ of $ M $ is defined as in the case of Riemannian manifolds. By definition, $ M $ is a maximal hypersurface if the mean curvature $ H $ of $ M $ is identically zero. S. Ishihara proved that a complete maximal space-like hypersurface $ M $ in $ H _ {1} ^ {n + 1 } ( c ) $ satisfies $ S \leq nc $, and $ S = nc $ if and only if $ M $ is isometric to the hyperbolic cylinder $ H ^ {k} ( c _ {1} ) \times H ^ {n - k } ( c _ {2} ) $, where $ S $ is the squared norm of the second fundamental form of $ M $ and $ H ^ {k} ( c _ {i} ) $, $ i = 1, 2 $, is a $ k $- dimensional hyperbolic space of constant curvature $ c _ {i} $. The rigidity of the hyperbolic cylinder $ H ^ {k} ( c _ {1} ) \times H ^ {n - k } ( c _ {2} ) $ in $ H _ {1} ^ {n + 1 } ( c ) $ was proved by U.-H. Ki, H.S. Kim and H. Nakagawa [a3]: for a given integer $ n $ and constant $ c > 0 $, there exists a constant $ C < nc $, depending on $ n $ and $ c $, such that the hyperbolic cylinder $ H ^ {k} ( c _ {1} ) \times H ^ {n - k } ( c _ {2} ) $ is the only complete maximal space-like hypersurface in $ H _ {1} ^ {n + 1 } ( c ) $ of constant scalar curvature and such that $ S > C $. In particular, for $ n = 3 $, Q.M. Cheng [a1] has characterized the complete maximal space-like hypersurfaces in $ H _ {1} ^ {4} ( c ) $ under the condition of constant Gauss–Kronecker curvature (cf. Gaussian curvature): Let $ M $ be a $ 3 $- dimensional complete maximal space-like hypersurface of $ H _ {1} ^ {4} ( c ) $. Now:

1) if the Gauss–Kronecker curvature of $ M $ is a non-zero constant, then $ M $ is the hyperbolic cylinder $ H ^ {1} ( c _ {1} ) \times H ^ {2} ( c _ {2} ) $;

2) if the scalar curvature $ K $ is constant and $ \inf K ^ {2} > 0 $, then $ M $ is the hyperbolic cylinder $ H ^ {1} ( c _ {1} ) \times H ^ {2} ( c _ {2} ) $. There are no complete maximal space-like hypersurfaces in $ H _ {1} ^ {4} ( c ) $ with constant scalar curvature and $ \sup K ^ {2} < { {S ^ {3} } / {54 } } $.

On the other hand, complete space-like submanifolds in anti-de Sitter spaces with parallel mean curvature have been investigated by many authors.

Cf. also De Sitter space.

References

[a1] Q.M. Cheng, "Complete maximal space-like hypersurfaces of $H_1^4(c)$" Manuscr. Math. , 82 (1994) pp. 149–160
[a2] T. Ishikawa, "Maximal space-like submanifolds of a pseudo–Riemannian space of constant curvature" Michigan Math. J. , 35 (1988) pp. 345–352
[a3] U-H. Ki, H.S. Kim, H. Nakagawa, "Complete maximal space-like hypersurfaces of an anti-de Sitter space" Kyungpook Math. J. , 31 (1991) pp. 131–141
How to Cite This Entry:
Anti-de Sitter space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Anti-de_Sitter_space&oldid=11698
This article was adapted from an original article by Qingming Cheng (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article