Geometric Analysis Applied to General Relativity

布雷教授研究与标量曲率有关的几何分析问题，其中许多问题都是由广义相对论中的基本问题引起的。 最近，Marcus Khuri和PI在时空的渐近平坦类空切片的彭罗斯猜想上取得了重要进展，他们将该猜想简化为某些自然动力系统的有趣的存在性问题。's. 其中一个存在性问题类似于Huisken-Ilmanen解决的证明具有跳跃的逆平均曲率流的存在性的问题，但是对于两个方程而不是一个方程的系统。 彭罗斯猜想的物理解释是一个自然的想法，即具有非负能量密度的时空的总质量至少应该是时空中黑洞贡献的质量。 在1973年，罗杰·彭罗斯能够将上述陈述转化为关于时空的渐近平坦类空切片（其本身是黎曼三维流形）的柯西数据的精确几何猜想。 时间对称的情况，被称为黎曼彭罗斯猜想，在1999年由PI证明，在1997年由Huisken-Ilmanen证明了单个黑洞。 在这种情况下，时空的能量密度等于切片的标量曲率，总质量是一个描述黎曼流形在无穷远处变平的速率的参数，黑洞的视视界是面积最小化的极小曲面。 黎曼彭罗斯不等式是这样的陈述，即总质量大于或等于黑洞视视界表面积的平方根除以16 π。 当我们放弃黎曼流形在时空中是时间对称的假设，并允许切片的第二基本形式是任何东西时，这个陈述的推广被简单地称为彭罗斯猜想。 PI还研究广义相对论中的负点质量奇点和与准局部质量有关的问题。作为广受赞誉的广义相对论理论，该理论的基本方面仍然没有得到理解。 例如，给定宇宙在（坐标）时间的一个瞬间的状态，目前还不知道相关的方程，包括爱因斯坦方程，是否可以在时间上向前求解，而不是在很短的时间内。 然而，正如我们每天所观察到的，宇宙是不间断地存在的。 因此，理解广义相对论中的柯西问题的存在性理论，因为这个问题被称为，是一个主要问题。 如果广义相对论没有一个物理存在理论，这将是一个关于如何修改理论的主要提示。 如果广义相对论确实有一个对应于物理宇宙的存在理论，那么这将是支持该理论的又一个证据。 有一件事是清楚的，黑洞和奇点在这个问题中扮演着重要的角色。 无论如何，理解广义相对论的基本理论问题不仅会加深我们对该理论的理解，而且有助于当前和未来的研究人员发展下一代物理理论。

Professor Bray studies geometric analysis problems that relate to scalar curvature, many of which are motivated by fundamental questions in General Relativity. Recently, Marcus Khuri and the PI have made important progress on the Penrose Conjecture for asymptotically-flat space-like slices of spacetimes by reducing the conjecture to interesting existence questions for certain naturally motivated systems of p.d.e.'s. One of these existence questions is similar to the one solved by Huisken-Ilmanen to prove the existence of the inverse mean curvature flow with jumps, but for a system of two equations instead of one equation. The physical interpretation of the Penrose Conjecture is the natural idea that the total mass of a spacetime with nonnegative energy density should be at least the mass contributed by the black holes in the spacetime. In 1973, Roger Penrose was able to turn the above statement into a precise geometric conjecture about the Cauchy data of an asymptotically flat space-like slice (itself a Riemannian 3-manifold) of a spacetime. The time-symmetric case, known as the Riemannian Penrose Conjecture, was proved by the PI in 1999, and by Huisken-Ilmanen in 1997 for a single black hole. In this case, the energy density of the spacetime equals the scalar curvature of the slice, the total mass is a parameter describing the rate at which the Riemannian manifold is becoming flat at infinity, and apparent horizons of black holes are area-outerminimizing minimal surfaces. The Riemannian Penrose Inequality is the statement that the total mass is greater than or equal to the square root of the surface area of the apparent horizons of the black holes divided by 16 pi. When we drop the assumption that the Riemannian manifold is time-symmetric in the spacetime and allow the second fundamental form of the slice to be anything, a generalization of this statement is known simply as the Penrose Conjecture. The PI also studies negative point mass singularities in General Relativity and questions relating to quasi-local mass.As acclaimed a theory as General Relativity is, fundamental aspects of the theory are still not understood. For example, given the state of the universe at one instant of (coordinate) time, it is not currently known if the relevant equations, including the Einstein equation, can be solved forward in time other than for a very short period. Yet the universe, as we observe daily, exists without interruption. Hence, understanding the existence theory of the Cauchy Problem in General Relativity, as this problem is called, is a major question. If General Relativity does not has a physical existence theory, this will be a major hint as to how the theory needs to be modified. If General Relativity does have an existence theory corresponding to the physical universe, then this will be one more piece of evidence supporting the theory. One thing that is clear is that black holes and singularities play an important role in this question. In any case, understanding fundamental theoretical questions about General Relativity will not only deepen our understanding of the theory, but also help current and future researchers develop the next generation of physical theories.