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.

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.