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Seminar

Geometry: The moduli space of 2-convex embedded spheres April 13, 2016 (11:00 AM PDT - 12:00 PM PDT)
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Location: MSRI: Simons Auditorium
Speaker(s) Robert Haslhofer (University of Toronto)
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The moduli space of 2-convex embedded spheres

 


We investigate the topology of the space of smoothly embedded n-spheres in R^{n+1}, i.e. the quotient space M_n:=Emb(S^n,R^{n+1})/Diff(S^n). By Hatcher’s proof of the Smale conjecture, M_2 is contractible. This is a highly nontrivial theorem generalizing in particular the Schoenflies theorem and Cerf’s theorem that the space of orientation preserving diffeomorphisms of S^3 is path-connected.

In this talk, I will explain how geometric analysis can be used to study the topology of M_n respectively some of its variants.

I will start by sketching a proof of Smale's classical result that M_1 is contractible. By a beautiful theorem of Grayson, the curve shortening flow deforms any closed embedded curve in the plane to a round circle, and thus gives a geometric analytic proof of the fact that M_1 is path-connected. By flowing, roughly speaking, all curves simultaneously, one can improve path-connectedness to contractibility.

In the second half of my talk, I’ll describe recent work on the space of smoothly embedded spheres in the 2-convex case, i.e. when the sum of the two smallest principal curvatures is positive. Our main theorem (joint with Buzano and Hershkovits) proves that this space is path-connected, for every n. The proof uses mean curvature flow with surgery.


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