Yamabe metrics on cylindrical manifolds
This is joint work with B. Botvinnik. We study a particular class of open
manifolds. In the smooth category these are manifolds with cylindrical
ends. We give a natural setting for conformal geometry on such manifolds
including an appropriate notion of cylindrical Yamabe constant/invariant.
This leads to a corresponding version of the Yamabe problem on cylindrical
manifolds. We affirmatively solve this Yamabe problem: we prove existence
results and analyze singularities of minimizing metrics near infinity.
These singularities turn out to be of a very particular type: either almost
conical or almost cusp singularities. We describe the limiting case,
i.e., when the cylindrical Yamabe constant is equal to the Yamabe invariant
of the sphere. We prove that in this case a cylindrical manifold coincides
with the standard sphere punctured at finite number of points. In the course
of studying the limiting case we establish a Positive Mass Theorem for
manifolds with conical singularities. As a by-product, we revisit known
results on surgery and the Yamabe invariant.
Fall
2001 meeting of the PNGS