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4:00 pm Wednesday, February 13, 2013 Geometry-Analysis Seminar: Critical metrics on connected sums of Einstein manifoldsby Jeff Viaclovsky (University of Wisconsin) in HB 227- ABSTRACT: I will discuss a gluing procedure designed to obtain critical metrics of quadratic Riemannian functionals on connected sums of certain Einstein four-manifolds. Briefly, start with two Einstein four-manifolds of positive scalar curvature which are "rigid". Using the Green's function for the conformal Laplacian, convert one of these into an asymptotically flat (AF) scalar-flat metric. A "naive" approximate critical metric is obtained by scaling down this AF metric, and attaching it to the other compact Einstein metric minus a small ball using cutoff functions to obtain a smooth metric on the connected sum. It turns out that this naive approximate metric is too rough, and must be refined in order to compute the leading term of the Kuranishi map. For the main existence result, we then use two well-known Einstein manifolds as building blocks: the Fubini-Study metric on $\mathbb{CP}^2$ and the product metric on $S^2 \times S^2$. Using these factors in various gluing configurations, a zero of the Kuranishi map is then found for a specific quadratic Riemannian functional on certain connected sums. The exact functional depends on the geometry of the factors, and also on the mass of the AF metric. This is joint work with Matt Gursky.
Submitted by hardt@rice.edu |