Spencer Kelly, University of Waterloo
Sobolev Spaces Over Compact Manifolds
The space of smooth sections of a vector bundle over a manifold is an infinite dimensional Fréchet Space, and thus many of the tools used in finite-dimensional geometry are rendered useless on this space. However, taking the completion of this space with respect to the Sobolev norm, we obtain a Banach space. What's even better is that in the $L^2$ case we obtain a Hilbert space. In this talk we will walk through different constructions of the$L^2$-Sobolev spaces of sections of a vector bundle over a compact manifold, and discuss the Sobolev embedding theorem. We will also work through some of the properties of differential operators on this space and, time permitting, we will finish with the Berger-Ebin decomposition for differential operators with injective symbol.
MC 5417