## Geometry Colloquium

Seminar information archive ～04/15｜Next seminar｜Future seminars 04/16～

Date, time & place | Friday 10:00 - 11:30 126Room #126 (Graduate School of Math. Sci. Bldg.) |
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### 2013/04/11

10:00-11:30 Room #122 (Graduate School of Math. Sci. Bldg.)

Critical metrics on connected sums of Einstein four-manifolds (ENGLISH)

**Jeff Viaclovsky**(University of Wisconsin)Critical metrics on connected sums of Einstein four-manifolds (ENGLISH)

[ Abstract ]

I will discuss a gluing procedure designed to obtain critical metrics of quadratic Riemannian functionals on connected sums of certain Einstein four-manifolds. 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 identifying the boundary of a large ball in the AF metric with the boundary of a small ball in the other compact Einstein metric, using cutoff functions to glue together the AF metric with a suitably scaled compact metric in order 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. The main application is an existence result using 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. Using certain quotients of $S^2 ¥times S^2$ as one of the gluing factors, several non-simply connected examples are also obtained. This is joint work with Matt Gursky.

I will discuss a gluing procedure designed to obtain critical metrics of quadratic Riemannian functionals on connected sums of certain Einstein four-manifolds. 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 identifying the boundary of a large ball in the AF metric with the boundary of a small ball in the other compact Einstein metric, using cutoff functions to glue together the AF metric with a suitably scaled compact metric in order 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. The main application is an existence result using 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. Using certain quotients of $S^2 ¥times S^2$ as one of the gluing factors, several non-simply connected examples are also obtained. This is joint work with Matt Gursky.