23.05.2013, 16:15 Uhr  –  Albert Einstein Institut Potsdam, Room 0.63
Seminar "Topics in Geometric Analysis"

Reto Müller and Neshan Wickramasekera

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.

 
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