Proof of the Riemannian Penrose Conjecture Using the Positive Mass Theorem
Proof of the Riemannian Penrose Conjecture Using the Positive Mass Theorem
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用正质量定理证明黎曼彭罗斯猜想
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发表时间：
1999
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通讯作者：
H. Bray
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作者：
H. Bray
我们通过定义一个新的度量流来证明riemanian Penrose猜想，这是Roger Penrose在1973年提出的猜想的一个重要例子。这种度量流停留在包含最小球的非负标量曲率的渐近平坦黎曼3流形的范围内。特别是，如果我们将黎曼3流形视为广义相对论背景下时空的完全测地线子流形，那么总面积为a的最外层最小球体对应于质量为pa /16的黑洞视界，标量曲率对应于每一点的局部能量密度，而度规在无穷远处变平的速率对应于总质量。黎曼彭罗斯猜想指出，具有非负标量曲率的渐近平坦3流形的总质量大于或等于黑洞贡献的质量。我们定义的度规流不断地将原始的3度规演变为史瓦西3度规，它代表了真空中的球对称黑洞。我们定义的流动使得最小球体的面积（向外流动），因此在流动的每个指标中由黑洞贡献的质量是恒定的，然后使用正质量定理来表明指标的总质量是不增加的。然后，由于总质量等于史瓦西度规中黑洞的质量，黎曼彭罗斯猜想随之而来。这个结果改进了Huisken和Ilmanen[25]的漂亮工作，他们使用表面的逆平均曲率流来表明总质量至少是最大黑洞所贡献的质量。在第1节和第2节中，我们提出了这个问题，讨论了黑洞的总质量和视界等重要量，并给出了黎曼3流形的正质量定理和彭罗斯猜想。在第3节中，我们给出
We prove the Riemannian Penrose conjecture, an important case of a conjecture made by Roger Penrose in 1973, by defining a new flow of metrics. This flow of metrics stays inside the class of asymptotically flat Riemannian 3-manifolds with nonnegative scalar curvature which contain minimal spheres. In particular, if we consider a Riemannian 3-manifold as a totally geodesic submanifold of a space-time in the context of general relativity, then outermost minimal spheres with total area A correspond to apparent horizons of black holes contributing a mass p A/16�, scalar curvature corresponds to local energy density at each point, and the rate at which the metric becomes flat at infinity corresponds to total mass. The Riemannian Penrose conjecture then states that the total mass of an asymptotically flat 3-manifold with nonnegative scalar curvature is greater than or equal to the mass contributed by the black holes. The flow of metrics we define continuously evolves the original 3-metric to a Schwarzschild 3-metric, which represents a spherically symmetric black hole in vacuum. We define the flow such that the area of the minimal spheres (which flow outward) and hence the mass contributed by the black holes in each of the metrics in the flow is constant, and then use the positive mass theorem to show that the total mass of the metrics is nonincreasing. Then since the total mass equals the mass of the black holes in a Schwarzschild metric, the Riemannian Penrose conjecture follows. This result improves upon the beautiful work of Huisken and Ilmanen [25], who used inverse mean curvature flows of surfaces to show that the total mass is at least the mass contributed by the largest black hole. In section 1 and 2, we motivate the problem, discuss important quantities like total mass and horizons of black holes, and state the positive mass theorem and the Penrose conjecture for Riemannian 3-manifolds. In section 3, we give