Event
Graduate Seminar
Blow-up in the Parabolic Scalar Curvature Equation
Dr. Brian Smith
17 Oct 2006, 17:15
Consider a manifold foliated by topological 2-spheres. Suppose that the
intrinsic geometry of the foliation spheres has been specified. We would
like to obtain a manifold of prescribed scalar curvature in a non-conformal
way by modifying the metric only in a direction transverse to the foliation
spheres. That is, we want to find a function u so that the metric
g=u2dr2+h
has the desired scalar curvature
R, where
r is the foliating function
and
h denotes the metric of the foliation spheres. If the area element of
h is expanding with increasing
r then this gives rise to a parabolic
equation for
u in which
r plays the role of a time variable. It is
easily seen by using the maximum principle that in many cases of physical
interest the solution blows up at some finite value of
r, say
r1. The
purpose of this talk is to discuss a situation in which blow-up occurs in
such a way that the metric can nonetheless be continuously extended up to
r1, which corresponds to a horizon.
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