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A mozaic of mathematical problems, Dmitri Burago.
This won't be a typical seminar lecture. Instead I'll give a number of mini-talks on very different topics. The only thing linking the topics together is that they have all been of interest to me in the past several years, and the most important part of the lecture will be the presentation of open problems. These will be formulated using only basic material at the level of graduate student written exams.
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Rigidity of generic singularities of mean curvature flow, Bill Minicozzi.
I will talk about joint work with Toby Colding and Tom Ilmanen on
singularities of mean curvature flow (MCF).
Self-shrinkers are special solutions that evolve by rescaling and model the
singularities. There are infinitely many in each dimension,
but the only generic are round cylinders. We prove that these are rigid
in a very strong sense:
Any other self-shrinker that is sufficiently close to one of them on a
large, but compact, set must
itself be a round cylinder.
To our knowledge, this is the first general rigidity theorem for
singularities
of a nonlinear geometric flow. We expect that the techniques and ideas
developed here have applications
to other flows.
Our results hold in all dimensions.
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Yamabe flow, its ancient solutions and singularity formation Natasa Sesum.
We will discuss conformally flat complete Yamabe flow and show
that in some cases we can give the precise description of singularity
profiles close to the extinction time of the solution. We will also talk
about a construction of new compact ancient solutions to the Yamabe flow.
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Dynamical stability and instability of Ricci-flat metrics Reto Mueller.
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.
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The inverse mean curvature flow for hypersurfaces with boundary, Thomas Marquardt.
We consider hypersurfaces with boundary which evolve in the
direction of the unit normal with speed equal to the reciprocal of the mean
curvature. The boundary condition is of Neumann type, i.e. the evolving
hypersurface moves along but stays perpendicular to a fixed supporting
hypersurface.
In the case where the supporting hypersurface is a convex cone we prove
long-time existence for star-shaped initial hypersurfaces of strictly
positive mean curvature.
In the general case, however, one can not expect the flow to exist for all
time. Therefore, we use a level-set approach together with a variational
formulation to prove the existence of weak solutions. Furthermore, we
indicate the existence of a monotone quantity which is the analog of the
Hawking mass for closed hypersurfaces.
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