I will discuss the connection between Sasaki-Einstein metrics and algebraic geometry in the guise of K-stability. In particular, I will give a differential geometric perspective on K-stability which arises from the Sasakian view point, and use K-stability to find infinitely many non-isometric Sasaki-Einstein metrics on the 5-sphere. This is joint work with G. Szekelyhidi.
We discuss the following recent result of the speaker. Suppose a closed 3-manifold M has scalar curvature at least 1, and has nontrivial second homotopy group, and is not covered by the cylinder (S^2)*R. Then the pi_2-systole of M (i.e. the minimal area in the second homotopy group) is bounded by a constant that is approximately 5.44pi. If we include quotients of cylinder into consideration, then the best upper bound is weakened to 8_pi. This shows a topological gap in the pi_2-systolic inequality. We will discuss the ideas behind this theorem, as well as the proof using Huisken and Ilmanen’s weak inverse mean curvature flow.