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Uniform doubling for abelian products with $\operatorname{SU}(2)$

Nathaniel Eldredge, Maria Gordina, Laurent Saloff-Coste

TL;DR

The authors prove uniform volume doubling for every Lie group that is a quotient of $\mathrm{SU}(2)\times\mathbb{R}^n$, in particular establishing this property for $\mathrm{U}(2)$. They reduce the problem to decoupled left-invariant metrics on $\mathrm{SU}(2)\times\mathbb{R}^3$ via Milnor-type bases and liftings, giving explicit two-sided ball-volume estimates $\mu_0(B_g(r)) \asymp \overline{V}_g(r)$ where $\overline{V}_g$ depends only on the quadruple of parameters $(a_1,a_2,a_3,d)$. From these estimates, they deduce a uniform scale-invariant Poincaré inequality and, consequently, uniform heat-kernel bounds and Weyl-type spectral information across all left-invariant metrics on the class of groups considered. The results extend to quotients of $\mathrm{SU}(2)\times\mathbb{R}^n$ and yield uniform analytic consequences for a wide family of noncompact and compact groups, with concrete implications for heat kernels on corresponding homogeneous spaces.

Abstract

We prove that the uniform doubling property holds for every Lie group which can be written as a quotient group of $\operatorname{SU}(2) \times \mathbb{R}^n$ for some $n$. In particular, this class includes the four-dimensional unitary group $\operatorname{U}(2)$. As this class contain non-compact as well as compact Lie groups, we discuss a number of analytic and spectral consequences for the corresponding heat kernels.

Uniform doubling for abelian products with $\operatorname{SU}(2)$

TL;DR

The authors prove uniform volume doubling for every Lie group that is a quotient of , in particular establishing this property for . They reduce the problem to decoupled left-invariant metrics on via Milnor-type bases and liftings, giving explicit two-sided ball-volume estimates where depends only on the quadruple of parameters . From these estimates, they deduce a uniform scale-invariant Poincaré inequality and, consequently, uniform heat-kernel bounds and Weyl-type spectral information across all left-invariant metrics on the class of groups considered. The results extend to quotients of and yield uniform analytic consequences for a wide family of noncompact and compact groups, with concrete implications for heat kernels on corresponding homogeneous spaces.

Abstract

We prove that the uniform doubling property holds for every Lie group which can be written as a quotient group of for some . In particular, this class includes the four-dimensional unitary group . As this class contain non-compact as well as compact Lie groups, we discuss a number of analytic and spectral consequences for the corresponding heat kernels.

Paper Structure

This paper contains 16 sections, 39 theorems, 131 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Uniform doubling for quotients and homogeneous spaces
  3. Decoupled metrics
  4. Standard Milnor bases and generalizations
  5. A lifting construction
  6. The bi-invariant reference metric
  7. Parametrization of decoupled metrics
  8. Statement of volume estimates
  9. Milnor basis identities in $\operatorname{SU}(2)$
  10. Closed form expressions
  11. Maurer--Cartan forms of smooth curves
  12. Exponential identities
  13. Coordinates
  14. Lower bounds
  15. Upper bounds
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $G$ be a Lie group such that $G \cong (\mathrm{SU}(2) \times \mathbb{R}^n) / H$ for some closed normal subgroup $H \leqslant \mathrm{SU}(2) \times \mathbb{R}^n$. Then $G$ is uniformly doubling, with a constant $D(G)$ that depends only on $n$. In particular, the unitary group $\mathrm{U}(2)$ is u

Figures (2)

  • Figure 1: The hexagon $\widetilde{H}(\mu^{\ast}, \nu^{\ast}, \xi^{\ast})$
  • Figure 2: The set $H$ for various values of the parameters $d, \mu^\ast, \nu^\ast, \xi^\ast$, in dark gray. The cylinder $\mathbb{S} \times \mathbb{R}$ is shown in light gray for reference. The portion of $H$ between the vertical dashed lines represents $H_\iota$. Note that in (a) and (b), the projection $\pi : \widetilde{H} \to H$ is injective, but in (c) it is not.

Theorems & Definitions (81)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Example 2.2
  • Remark 2.3
  • Lemma 2.4: Guivarch1973a
  • proof
  • Theorem 2.5
  • proof
  • Corollary 2.6
  • ...and 71 more