Appearance of stable minimal spheres along the Ricci flow in positive scalar curvature
Student No.:50
Time:July 26 Wednesday, Jing Zhai 304, 3:00 pm.
Instructor:Antoine Song  
Place:Jing Zhai 304
Starting Date:2017-7-26
Ending Date:2017-7-26

Abstract: By a well-known theorem of Hamilton, if a closed 3-manifold M has positive Ricci curvature, then it remains so along the Ricci flow and the metric converges to a round metric after renormalization. Two immediate properties following from Ric>0 are that the scalar curvature is positive and that there are no stable minimal spheres. The positivity of the scalar curvature is preserved by the Ricci flow but is it the case for the other property? In other words: suppose that M has positive scalar curvature at time 0, can stable minimal spheres appear along the Ricci flow if there were none 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. As we will see, small stable minimal spheres are also closely related the formation of Type I singularities.