I will explain that to every small and decaying solution of the linearized constraint equations about Minkowski initial data, one can add a quadratically small correction to obtain a solution of the full constraint equations. Near spacelike infinity, the correction is given by Kerr black hole initial data, up to a term that decays faster than the linearized solution, and that has Schwartz decay if the linearized solution has Schwartz decay. The main tool is a right inverse (up to necessary integrability conditions) for the linearized constraint operator about Minkowski initial data, that has optimal mapping properties relative to weighted b-Sobolev spaces, where the weights measure decay towards infinity. Using a recent result, one obtains that the solutions of the Einstein equations with these initial data admit a regular conformal compactification along null and timelike infinity.

Hamilton's pinching conjecture, that three-dimensional complete non-compact manifolds with pinched Ricci curvature are flat, was resolved recently using Ricci flow. In this talk I will explain a direct analogue of that result in all dimensions. One aspect of the work is a new lifting technique that allows us to handle manifolds that are collapsed at infinity; until now this could only be achieved in 3D via work of Lott. The new theorem builds on earlier work of Deruelle, Simon and the speaker and separately of Lee-Topping. Joint work with Alix Deruelle, Man Chun Lee, Miles Simon and Peter Topping.

I will present a sharp quantitative version of the Alexandrov theorem on closed hypersurfaces with constant mean curvature, with applications to the analysis of the asymptotic behavior of the volume preserving and the Mullins-Sekerka flat flows.

The isometry group of the classical Lawson embedded minimal surface ξ_{2,1} ⊂ S³ of genus 2 is isomorphic to the group D₄ × S₃, where D₄ is the dihedral group of order 8 and S₃ the permutation group of order 6. Iso(ξ_{2,1}) has a subgroup of index 3 isomorphic to the bidihedral group D_{4h} = Z₂ × D₄. We will explain how to prove that ξ_{2,1} is the unique closed embedded minimal surface of genus 2 in S³ whose isometry group contains D_{4h}. This is a joint work with J. Pérez