ABSTRACT

The problem of the existence of a constant scalar curvature Kahler
metric in a given Kahler class is an important and difficult problem
and has provided the motivation for much current research in Kahler
geometry. For the special case of Kahler-Einstein metrics on Fano
manifolds, existence was conjectured by Yau to be equivalent to the
stability of the manifold in the sense of geometric invariant theory.
The principal investigator proposes to study three parabolic flows of
Kahler potentials which arise naturally in this context. The first is
the J-flow, which is the gradient flow of a functional appearing in
Chen's formula for the Mabuchi energy. The study of the J-flow has led
to significant advances in understanding the lower boundedness and
asymptotics of the Mabuchi energy. The second is the Kahler-Ricci
flow. Its behavior in the Fano case is not yet well understood, and it
is proposed that the method of multiplier ideal sheaves may capture the
necessary information about its singularities to be able to provide a
link with stability. The third is the Calabi flow. It is a fourth
order parabolic PDE about which little is known in general. The
principal investigator intends to study the problem of long time
existence of this flow.


An important problem in geometry and physics is whether a given space
has a special notion of distance. Take, for example, the two
dimensional sphere - the surface of a ball. With our usual sense of
distance, this space is curved in the same way at every point. We say
that the sphere admits a 'metric of constant curvature'. Not all spaces
admit such metrics, and it is an interesting and deep problem to find
conditions under which they do. A natural and beautiful approach to
this problem is the parabolic, or 'heat flow' method. The idea is
simple. The distribution of heat in an object (not subject to outside
sources) will flow in time, becoming more even and finally constant -
no matter what the initial distribution looked like. In a similar way,
if we start with an arbitrary notion of distance on a space, then we
can apply a natural 'heat flow' and hope to prove, under the right
conditions, that we obtain convergence to a metric of constant
curvature, or some other special metric, as time evolves. If no such
metrics exist, then we expect the flow to go wrong - to develop
singularities. The PI intends to study the question of convergence and
singularities of three such parabolic flows corresponding to different
types of special metrics.

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