Differential Geometry Seminar: Appearance of stable spheres along the Ricci flow in positive scalar curvature
Seminar | April 3 | 1:10-2 p.m. | 939 Evans Hall
Antoine Song, Princeton
For a 3-manifold M not isometric to the round sphere, with scalar curvature at least 6 and positive Ricci curvature, Marques and Neves proved a 3-dimensional version of the Toponogov theorem: there exists an embedded minimal surface S of area less than $4\pi $. Their proof uses a combination of min-max theory for minimal surfaces and the Ricci flow. While the general case (no assumption on the Ricci curvature) can now be proved with a rather different approach, it is interesting to ask to what extent this Ricci flow method can be extended to positive scalar curvature. A natural question arises: suppose that the scalar curvature is positive at time 0, can stable surfaces appear along the Ricci flow if there were none of them at time 0? I will show examples where not only stable spheres appear but a non-trivial singularity occurs. However, under suitable symmetry assumptions, this cannot happen. Small stable spheres are also closely related to Type I singularities. Consequences of the previous examples for the study of minimal surfaces will be discussed.