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

杰夫·A·维亚克洛夫斯基的项目包括三个部分。第一部分是与钢田的合作，研究了四维Bach平坦度量的奥布朗德紧性。第二部分是与Matt Gursky的合作，研究了三维曲率泛函和共形变形。第三部分讨论了共形几何中一些完全非线性方程的解的存在性。第一个项目是了解四维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，有许多已知的解，例如局部共形Einsten的度量，以及自对偶或反自对偶度量。由于已知这些度量的大量存在，因此理解解的模空间是一个有趣的问题。Viaclovsky和Tian的计划是证明，在一定的几何条件下，人们可以通过添加具有轨道奇点的度量来紧致这个空间。该项目的其他部分处理的是非保角几何中的完全非线性方程。其目的是使度量共形变形，使得Schouten张量的特征值的第k个初等对称函数是常数。这可以看作是著名的Yamabe问题的完全非线性推广。它的一个应用是通过保角变形来改善曲率。例如，给定一个初始度量，找到使该度量可以共形变形为正截面曲率的条件。与田的项目推广了爱因斯坦度规的一些众所周知的结果。爱因斯坦度规特别有趣，因为它与爱因斯坦的广义相对论相联系，而巴赫方程是爱因斯坦方程的自然高阶推广。共形几何中的工作自然是共形不变的，通过共形场理论在物理中也有应用。此外，这些方程出现了新的质量概念，这也是广义相对论的兴趣所在。这些方程还可用于确定3-流形的拓扑，只要给出曲率的一些几何约束。

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.