Scalar Curvature, Geometric Flow, and the General Penrose Conjecture

标量曲率、几何流和广义彭罗斯猜想（Penrose Conjecture）这项研究的主要目标是证明广义相对论中关于时空中黑洞质量的完整彭罗斯猜想。在3+1维空间中，这一猜想表明，一个处处具有非负能量密度的时空的总质量大于或等于该时空中所有黑洞视界总面积的平方根除以16。这个猜想可以被认为是说，一群黑洞所贡献的质量至少是它们表面积的平方根除以16，所以宇宙中其他地方的非负能量密度迫使时空的总质量至少是这个量。彭罗斯猜想最好被认为是一个关于任意三维类空间时空切片的猜想。在假定类空间片具有零秒基本形式的特殊情况下，这个猜想被称为黎曼彭罗斯猜想。这个猜想最初是由Huisken和Ilmanen在1997年为单个黑洞证明的，然后作者在1999年为任意数量的黑洞证明了这个猜想。作者还想在大于3维的情况下证明黎曼彭罗斯猜想，并即将在小于8维的情况下宣布这一结果。由于黑洞的视界表现为极小的超表面，可以具有协维7的奇点，因此第八维及更高的维度提出了额外的几何和分析挑战。由于类似的原因，正质量定理在这些维度上仍然是开放的，这是另一个非常有趣的相关问题。提出上述问题的动机是为了更好地理解广义相对论。虽然爱因斯坦的广义相对论在实验上是最著名的引力理论，但关于这一理论的许多理论问题根本没有得到很好的理解。例如，给定宇宙在某个初始时刻，爱因斯坦方程是否如人们所希望的那样在未来有唯一的、表现良好的解，或者奇点是否会出现，例如，可能会辐射或消耗能量？该理论还预测了黑洞的存在，天文学家相信他们已经能够探测到许多黑洞的位置，其中包括位于银河系中心的一个质量是太阳300万倍的黑洞。由于目前对黑洞进行实验是不可行的，因此从理论上理解这一潜在的非常重要的现象是很有必要的。这个研究项目希望能更好地理解黑洞以及广义相对论中的能量和质量。

Scalar Curvature, Geometric Flows, and the General Penrose Conjecture DMS - 0206483 Hubert Bray, MIT The primary goal of this research is to prove the full Penrose conjecture in general relativity about the mass of black holes in a spacetime. In 3+1 dimensions, this conjecture states that the total mass of a spacetime with nonnegative energy density everywhere is greater than or equal to the square root of the total area of the event horizons of all of the black holes in the spacetime divided by 16 pi. This conjecture can be thought of as stating that the mass contributed by a collection of black holes is at least the square root of their surface areas divided 16 pi, so that nonnegative energy density everywhere else in the universe forces the total mass of the spacetime to be at least this amount. The Penrose conjecture is best thought of as a conjecture on arbitrary three dimensional space-like slices of the spacetime. In the special case that the space-like slice is assumed to have zero second fundamental form, the conjecture is known as the Riemannian Penrose conjecture. This conjecture was first proved for a single black hole by Huisken and Ilmanen in 1997 and then for any number of black holes by the author in 1999. The author would also like to prove the Riemannian Penrose Conjecture in dimensions higher than three and is close to announcing this result for dimensions less than eight. Dimensions eight and higher present additional geometric and analytical challenges arising from the fact that the apparent horizons of black holes manifest themselves as minimal hypersurfaces which can have co-dimension seven singularities. The positive mass theorem is also still open in these dimensions for similar reasons, and is another very interesting related problem.The motivation for the above problems is to gain a better understanding of General Relativity. While Einstein's theory ofGeneral Relativity is experimentally the best known theory of gravity, their are many theoretical questions about the theory which are not well understood at all. For example, given theuniverse at some initial time, do the Einstein equations have unique well-behaved solutions in the future as one would hope, ordo singularities typically occur which might radiate or consumeenergy for example? The theory also predicts the existence ofblack holes, and astronomers believe that they have been able to detect the location of many black holes including one which is 3 million times the mass of the sun at the center of our galaxy. Since doing experiments with black holes is currently not feasible, it makes since to understand this potentially very important phenomenon on a theoretical level for now. This research project hopes to lead to a better understanding of blackholes as well as energy and mass in General Relativity.