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.

Ossian Bonnet (1819–1892) classified the surfaces in Euclidean 3-space that can be isometrically deformed without changing the mean curvature function H, showing that there are two types: the surfaces of constant mean curvature and a 4-dimensional ‘exceptional family’ (with variable mean curvature) that are now known as Bonnet surfaces. The corresponding problem in affine 3-space is much more difficult, and the full classification is still unknown. More than 10 years ago, I classified the affine surfaces that can isometrically deformed (with respect to the induced Blaschke metric) while preserving their affine mean curvature in a 3-dimensional family (the maximum dimension possible), showing that they depend on 2 functions of 1 variable in Cartan’s sense. When I gave a talk* in this seminar about these results on September 10, 2013, I only knew that these surfaces corresponded to pseudoholomorphic curves in a certain almost-complex surface. However, I have recently shown that the structure equations for these mysterious surfaces can be interpreted as describing holomorphic Legendrian curves in CP^3 subject to a natural positivity condition, and the integration corresponds to a flat sp(2,R) connection, i.e., they can be interpreted as a Lax pair, but of a very special kind, for which the integration can be effected explicitly. I’ll explain these results and use them to show how the classical problem of determining the affine surfaces with constant affine mean curvature and constant Gauss curvature of the Blaschke metric can be explicitly integrated, which, heretofore, was unknown. * https://www4.math.duke.edu/media/watch_video.php?v=6948e657e69cadbaa1a6915335e9ea87