School of Mathematics

Geometry of 2-dimensional Riemannian disks and spheres.

Regina Rotman
University of Toronto; Member, School of Mathematics
March 11, 2019

I will discuss some geometric inequalities that hold on
Riemannian 2-disks and 2-spheres. 

For example, I will prove that on any Riemannian 2-sphere there M exist
at least three simple periodic geodesics of length at most 20d, where d is the diameter of M, (joint with A. Nabutovsky, Y. Liokumovich).
This is a quantitative version of the well-known Lyusternik and Shnirelman theorem.

Filling metric spaces

Alexander Nabutovsky
University of Toronto; Member, School of Mathematics
March 8, 2019

Abstract:  Uryson k-width of a metric space X measures how close X is to being k-dimensional. Several years ago Larry Guth proved that if M is a closed n-dimensional manifold, and the volume of each ball of radius 1 in M does not exceed a certain small

constant e(n), then the Uryson (n-1)-width of M is less than 1. This result is a significant generalization of the famous Gromov's inequality relating the volume and the filling radius that plays a central role in systolic geometry.

 

Normalized harmonic map flow

Michael Struwe
ETH Zürich
March 6, 2019

Abstract: Finding non-constant harmonic 3-spheres for a closed target manifold N is a prototype of a super-critical variational problem. In fact, the 
direct method fails, as the infimum of the Dirichlet energy in any homotopy class of maps from the 3-sphere to any closed N is zero; moreover, the 
harmonic map heat flow may blow up in finite time, and even the identity map from the 3-sphere to itself is not stable under this flow. 

Existence and uniqueness of Green's function to a nonlinear Yamabe problem

Yanyan Li
Rutgers University
March 6, 2019

Abstract: For a given finite subset S of a compact Riemannian manifold (M; g) whose Schouten curvature tensor belongs to a given cone, we establish a necessary and
sufficient condition for the existence and uniqueness of a conformal metric on $M \setminus S$ such that each point of S corresponds to an asymptotically flat end and
that the Schouten tensor of the new conformal metric belongs to the boundary of the given cone.  This is a joint work with Luc Nguyen.