Appearance of stable spheres along the Ricci flow in positive scalar curvature

Appearance of stable spheres along the Ricci flow in positive scalar curvature

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Antoine Song, Princeton University
Fine Hall 314

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 4pi. 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 desirable to extend the Ricci flow method, for it yields more geometric information on S. When trying to do so, 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. Stable spheres are also related to Type I singularities. The previous examples motivate the definition of a certain "local min-max", useful when generalizing the min-max/Ricci flow method to manifolds with positive scalar curvature only.