16:15 Uhr | Reto Müller (Imperial College) | Dynamical stability and instability of Ricci-flat metrics
Let $M$ be a compact manifold. A Ricci-flat metric on $M$ is a
Riemannian metric with vanishing Ricci curvature. Ricci-flat metrics are
fairly hard to construct, and their properties are of great interest. They are
the critical points of the Einstein-Hilbert functional, the fixed points of
Hamilton’s Ricci flow and the critical points of Perelman’s $\lambda$-functional.
In this talk, we are concerned with the stability properties of Ricci-flat
metrics under Ricci flow. We will explain the following stability and
instability results. If a Ricci-flat metric is a local maximizer of $\lambda$,
then every Ricci flow starting close to it exists for all times and converges
(modulo diffeomorphisms) to a nearby Ricci-flat metric. If a Ricci-flat metric
is not a local maximizer of $\lambda$, then there exists a nontrivial ancient
Ricci flow emerging from it. This is joint work with Robert Haslhofer.
|
17:45 Uhr | Neshan Wickramasekera (Cambridge) | A sharp strong maximum principle for singular minimal hypersurfaces
If two smooth, connected, embedded minimal hypersurfaces with no
singularities satisfy the property that locally near every common point
$p$, one hypersurface lies on one side of the other, then it is a direct
consequence of the Hopf maximum principle that either the hypersurfaces
are disjoint or they coincide. It is a natural question to ask if the
same result must extend to pairs of singular minimal hypersurfaces
(stationary codimesion 1 integral varifolds) with connected supports;
in this case the above one hypersurface lies locally on one side of the
other hypothesis can naturally be imposed for each common point $p$
which is a regular point of at least one hypersurface.
The answer to this question in general is no in view of simple examples
such as two pairs of transversely interecting hyperplanes with a common
axis. The answer however is yes
if the singular set of one of the hypersurfaces has $(n-1)$-dimesional
Hausdorff measure zero, where $n$ is the dimension of the hypersurfaces.
I will discuss this result, which generalizes and unifies the previous
strong
maximum principles of Ilmanen and Solomon-White.
|