Compactness of Critical Metrics and Some Fully Nonlinear Equations in Conformal Geometry

摘要DMS - 0202477.Jeff A. Viaclovsky的项目有三个部分。第一部分是与田刚的合作，研究了4维Bach平坦度规的轨道紧性。第二部分是与Matt Gursky的联合工作，是关于3维曲率的曲率泛函和共形变形。第三部分讨论了共形几何中一类完全非线性方程解的存在性。 第一个项目是理解四维Weyl张量的L2范数临界点的模空间的紧性。欧拉-拉格朗日方程被称为巴赫方程，并且有许多已知的解，例如，局部共形的爱因斯坦度量，以及自对偶或反自对偶度量。由于已知这些度量大量存在，理解解的模空间是一个有趣的问题。该项目的Viaclovsky和田是表明，在一定的几何条件下，人们可以紧这个空间增加度量与orbifold奇点。该项目的其他部分处理完全非线性方程inconformal几何。我们的目标是共形变形的度量，使第k个基本对称函数的特征值的Schouten张量是常数。这可以看作是一个完全非线性推广的著名Yamabe问题。这一点的一个应用是通过保形变形来改善曲率。例如，给定一个初始度量，条件被发现，使度量可以共形变形为正截面曲率。田的项目推广了爱因斯坦度量的一些著名结果。爱因斯坦度量与爱因斯坦的广义相对论的联系特别有趣，巴赫方程是爱因斯坦方程的自然高阶推广。在共形几何中的工作是自然的共形不变的，并且通过共形场论在物理学中也有应用。首先，这些方程中出现了新的质量概念，这也是广义相对论感兴趣的。这些方程也有应用，以确定拓扑的3流形，给定一些几何约束的曲率。

ABSTRACT DMS - 0202477.Jeff A. Viaclovsky's project has three parts. The first part is joint work with Gang Tian, dealing with orbifold compactness of Bach flat metrics in dimension 4. The second part is joint work with Matt Gursky, and is about curvature functionals and conformal deformation of curvatures in dimension 3. The third part deals with existence of solutions to some fully nonlinear equations in conformal geometry. The first project is to understand compactness properties of the moduli space of critical points of the L2 norm of the Weyl tensor in 4 dimensions. The Euler-Lagrange equations are known as the Bach equations, and there are many known solutions, for example, metrics that are locally conformally Einsten, and self-dual or anti-self-dual metrics. Since it is known that these metrics exist in abundance, it is an interesting problem to understand the moduli space of solutions. The project of Viaclovsky and Tian is to show that, with certain geometric conditions, one may compactify this space by adding metrics with orbifold singularities. The other parts of the project deal with fully nonlinear equations inconformal geometry. The goal is to conformally deform a metric so that the kth elementary symmetric function of the eigenvalues of the Schouten tensor is constant. This may be viewed as a fully nonlinear generalization of the well-known Yamabe problem. An application of this is to improve the curvature by conformal deformation. For example, given an initial metric, conditions are found so that the metric can be conformally deformed to positive sectional curvature. The project with Tian generalizes some well-known results for Einstein metrics. Einstein metrics are particularly interesting from the connection with Eisteins's theory of general relativity, and the Bach equations are a natural higher order generalization of the Einstein equation. The work in conformal geometry is naturally conformally invariant and also has applications in physics through conformal field theory. Firthermore, new notions of mass arise for these equations, and this is also of interest in general relativity. These equations also have applications to determining the topology of 3-manifolds, given some geometric constraint on the curvature.