Evolution Equations in Geometry (I)

Evolution equations have been used to address successfully key questions in Differential Geometry
like isoperimetric inequalities, the Poincaré conjecture, Thurston’s geometrization conjecture, or
the differentiable sphere theorem.

During this series of lectures we will give a general introduction to geometric flows, which are sort of
non-linear versions of the heat equation for a relevant geometric quantity. These equations should be
understood as a tool to canonically deform a manifold into a manifold with nicer properties, for instance,
some kind of constant curvature. We will focus on the mean curvature flow and the Ricci flow. We will
learn some basic tools of the theory including short time existence, derivation of the evolution equations,
maximum principles, etc.

In order to control the behaviour of the flow, we will look for properties of the manifold that are preserved
under the flow. This is typically the case for a large family of non-negative curvature conditions. In contrast,
the condition of almost non-negative curvature operator (e.g. the condition that its smallest eigenvalue is
larger than -1) is not preserved under Ricci flow. In the last talk I will present a recent work (joint with
Richard Bamler and Burkhard Wilking) showing that, however, that a metric whose curvature operator has
eigenvalues larger than -1 can be evolved to a Ricci flow on a time-interval of uniform size. Moreover,
the eigenvalues of the curvature operator on this flow remain at least bounded from below by a uniform
negative constant. We will outline the proof of this result and discuss applications.