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

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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.