Abstract:
The Laplacian on a compact Riemannian manifold enjoys a wide variety of functional inequalities: spectral gap, Poincar’e inequality, parabolic Harnack inequality, heat kernel estimates, and more. When the Riemannian manifold is a compact Lie group with a left-invariant Riemannian metric, it can be shown that all these functional inequalities follow from the simple property of volume doubling: doubling the radius of a ball increases its volume by at most a constant factor (the doubling constant of the metric). Motivated by this, we have shown that on the compact Lie group SU(2), there is a uniform bound on the doubling constants of all left-invariant Riemannian metrics. This leads to functional inequalities holding uniformly over all such metrics, despite the absence of uniform curvature bounds.
This is joint work with Maria Gordina and Laurent Saloff-Coste.