Abstract:

On compact manifold, differentiable sphere theorem infers that manifold with 1/4 pinched positive curvature are necessarily sphere. Using Ricci flow method, it was shown by Brendle-Schoen that indeed the sphere theorem holds under an even weaker pinched 1-isotropic curvature condition. Motivated by this, It was conjectured by Hamilton-Lott that the in three manifold, noncompact manifold with pinched nonnegative Ricci (equivalent to 1-isotropic curvature) is necessarily flat. In this talk, we will discuss some recent advances in three dimensions and its generalization in higher dimension. This is based on joint work with P. Topping.