Left-invariant geometries on $\mathrm{SU}(2)$ are uniformly doubling

TL;DR

This paper proves that all left-invariant geometries on the group SU(2) have a uniform doubling property, providing volume estimates and implications for spectral analysis and heat kernels.

Contribution

It establishes the uniform doubling property for all left-invariant geometries on SU(2) and offers detailed volume estimates applicable to any radius.

Findings

Left-invariant geometries on SU(2) are uniformly doubling.

Provides explicit volume estimates for balls in these geometries.

Discusses implications for Laplacian spectra and heat kernels.

Abstract

A classical aspect of Riemannian geometry is the study of estimates that hold uniformly over some class of metrics. The best known examples are eigenvalue bounds under curvature assumptions. In this paper, we study the family of all left-invariant geometries on $\mathrm{SU}(2)$. We show that left-invariant geometries on $\mathrm{SU}(2)$ are uniformly doubling and give a detailed estimate of the volume of balls that is valid for any of these geometries and any radius. We discuss a number of consequences concerning the spectrum of the associated Laplacians and the corresponding heat kernels.