Speaker: Professor Nate Eldredge, University of Northern Colorado

Title: Uniform Volume Doubling for Compact Lie Groups

Abstract

The volume doubling constant of a Riemannian manifold has a simple definition: if you double the radius of a metric ball, by what factor does its volume increase? The answer is 2^n for flat n-dimensional Euclidean space, but can be larger in manifolds with negative curvature. The doubling constant turns out to be an important measure of the “badness” of a Riemannian manifold, and is especially informative in the presence of additional symmetry, such as a compact Lie group with a left-invariant metric. In this context, the doubling constant controls many other quantities of interest in geometric analysis, such as the Laplacian spectral gap, which are also classically controlled by Ricci curvature lower bounds.

We show that for the three-dimensional compact Lie group SU(2) of 2×2 unitary complex matrices with determinant +1, there is a uniform upper bound on the doubling constants of all left-invariant Riemannian metrics. In other words, all such metrics are uniformly doubling; they are all “comparable” to each other in this sense, and cannot be arbitrarily “bad.” This is particularly interesting because in the more traditional sense of Ricci curvature, they can be very different, and can be arbitrarily “bad” (negatively curved).

I will introduce some background and context for this result, some of its implications, and a few of the ideas in the proof, as well as a conjecture that the same property may hold on every compact connected Lie group.

This is joint work with Maria Gordina (University of Connecticut) and Laurent Saloff-Coste (Cornell University).

Note: Colloquium date was changed from February 1, 2019 to February 8, 2019.