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.
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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. |