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Rigidity of the subelliptic heat kernel on $\operatorname{SU}(2)$

Maria Gordina, Jing Wang

TL;DR

This work characterizes a sub-Riemannian heat-kernel rigidity phenomenon on $SU(2)$ by linking diffusion data to global geometric structure. The authors construct two interlocking isometries, $\Phi:(M,\delta)\to(\mathcal{S}^3)$ and $\Psi:(\mathbf{B},r)\to(\mathcal{S}^2)$, via a metric $\delta$ built from pseudo-metrics $r$ and $\theta$ and through a quotient by fibre leaves, and they analyze associated heat kernels $q_t$ and $\tilde{q}_t$ to obtain a Riemannian submersion with totally geodesic fibres. The main result is a bundle isometry between the given space and the Hopf fibration $S^1\to SU(2)\to \mathbb{CP}^1$, i.e., the sub-Riemannian sphere on $SU(2)$, showing that an explicit heat-kernel form rigidly determines the underlying sub-Riemannian geometry. The approach blends Dirichlet-form theory, spectral decomposition, and disintegration along fibres to realize a global geometric identification from diffusion data, with potential implications for shape analysis in sub-Riemannian contexts. Overall, the paper bridges diffusion-based rigidity with Hopf-fibration geometry, providing a concrete sub-Riemannian rigidity paradigm for $SU(2)$.

Abstract

We study heat kernel rigidity for the Lie group $\operatorname{SU}\left( 2 \right)$ kernel equipped with a sub-Riemannian structure. We prove that a metric measure space equipped with a heat kernel of a special form is bundle-isometric to the Hopf fibration $\operatorname{U}\left( 1 \right)\to \operatorname{SU}\left( 2 \right)\to \mathbb{CP}^1$, which coincides with the sub-Riemannian sphere $\operatorname{SU}\left( 2 \right)$.

Rigidity of the subelliptic heat kernel on $\operatorname{SU}(2)$

TL;DR

This work characterizes a sub-Riemannian heat-kernel rigidity phenomenon on by linking diffusion data to global geometric structure. The authors construct two interlocking isometries, and , via a metric built from pseudo-metrics and and through a quotient by fibre leaves, and they analyze associated heat kernels and to obtain a Riemannian submersion with totally geodesic fibres. The main result is a bundle isometry between the given space and the Hopf fibration , i.e., the sub-Riemannian sphere on , showing that an explicit heat-kernel form rigidly determines the underlying sub-Riemannian geometry. The approach blends Dirichlet-form theory, spectral decomposition, and disintegration along fibres to realize a global geometric identification from diffusion data, with potential implications for shape analysis in sub-Riemannian contexts. Overall, the paper bridges diffusion-based rigidity with Hopf-fibration geometry, providing a concrete sub-Riemannian rigidity paradigm for .

Abstract

We study heat kernel rigidity for the Lie group kernel equipped with a sub-Riemannian structure. We prove that a metric measure space equipped with a heat kernel of a special form is bundle-isometric to the Hopf fibration , which coincides with the sub-Riemannian sphere .
Paper Structure (14 sections, 33 theorems, 224 equations)

This paper contains 14 sections, 33 theorems, 224 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Dirichlet forms, semigroups and heat kernels
  4. Assumptions
  5. The first isometry
  6. The metric space $(M,\delta)$
  7. Dirichlet form and heat kernel on $(M, \delta, \mu)$
  8. Volume growth and dimension estimate of $(M,\delta,\mu)$
  9. The first isometry
  10. The second isometry
  11. The quotient space $(\mathbf{B},r,\tilde{\mu})$
  12. Heat kernel and Dirichlet form on $(\mathbf{B}, {r})$
  13. The second isometry
  14. Riemannian submersion and bundle isometry

Key Result

Proposition 2.6

Suppose $\left( M, r, \theta, \mu \right)$ satisfies Assumption Ass.HeatKernel. Then the measure $\mu$ is a probability measure, and the heat kernel $p_{t}\left( x, y \right)$ is in $L^{2}\left( M \times M, \mu \times \mu \right)$.

Theorems & Definitions (76)

  • Definition 2.1: Abstract heat kernels
  • Definition 2.2
  • Proposition 2.6: Properties of the heat kernel
  • proof
  • Lemma 2.7
  • proof
  • Remark 2.8
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • ...and 66 more