An Alexandrov–Fenchel-Type Inequality in Hyperbolic Space with an Application to a Penrose Inequality
An Alexandrov–Fenchel-Type Inequality in Hyperbolic Space with an Application to a Penrose Inequality
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DOI：
10.1007/s00023-015-0414-0
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发表时间：
2012-09
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影响因子：
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通讯作者：
L. L. Lima-L.;Frederico Girão
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作者：
L. L. Lima-L.;Frederico Girão
证明了n≥ 3的双曲空间中星形严格平均凸超曲面的一个Alexandrov-Fenchel型不等式.该参数使用了两个新的单调量沿着反平均曲率流。作为一个应用，我们建立了任意维上带有极小视界的渐近双曲图的最优Penrose不等式，当且仅当该图是反de Sitter-Schwarzschild解时，该不等式成立.这一结果改进了Dahl-Gicquaud-Sakovich的结果，并解决了这类初始数据集下具有负宇宙常数的时间对称时空的约束Penrose不等式.我们还解释了如何我们的方法可以很容易地适用于任何维数n ≥ 3的渐近局部双曲图的最优Penrose不等式。当视界具有亏格至少为1的紧致曲面的拓扑时，对于这类初始数据集，这为Gibbons、Chruelciel和Simon提出的关于奇异黑洞的Penrose型不等式的有效性的问题提供了肯定的答案。
We prove a sharp Alexandrov–Fenchel-type inequality for star-shaped, strictly mean convex hypersurfaces in hyperbolicn-space,n≥ 3. The argument uses two new monotone quantities along the inverse mean curvature flow. As an application we establish, in any dimension, an optimal Penrose inequality for asymptotically hyperbolic graphs carrying a minimal horizon, with the equality occurring if and only if the graph is an anti-de Sitter–Schwarzschild solution. This sharpens previous results by Dahl–Gicquaud–Sakovich and settles, for this class of initial data sets, the conjectured Penrose inequality for time-symmetric space–times with negative cosmological constant. We also explain how our methods can be easily adapted to derive an optimal Penrose inequality for asymptotically locally hyperbolic graphs in any dimensionn≥ 3. When the horizon has the topology of a compact surface of genus at least one, this provides an affirmative answer, for this class of initial data sets, to a question posed by Gibbons, Chruściel and Simon on the validity of a Penrose-type inequality for exotic black holes.