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Relative heat content asymptotics for sub-Riemannian manifolds

Andrei Agrachev, Luca Rizzi, Tommaso Rossi

Abstract

The relative heat content associated with a subset $Ω\subset M$ of a sub-Riemannian manifold, is defined as the total amount of heat contained in $Ω$ at time $t$, with uniform initial condition on $Ω$, allowing the heat to flow outside the domain. In this work, we obtain a fourth-order asymptotic expansion in square root of $t$ of the relative heat content associated with relatively compact non-characteristic domains. Compared to the classical heat content that we studied in [Rizzi, Rossi - J. Math. Pur. Appl., 2021], several difficulties emerge due to the absence of Dirichlet conditions at the boundary of the domain. To overcome this lack of information, we combine a rough asymptotic for the temperature function at the boundary, coupled with stochastic completeness of the heat semi-group. Our technique applies to any (possibly rank-varying) sub-Riemannian manifold that is globally doubling and satisfies a global weak Poincaré inequality, including in particular sub-Riemannian structures on compact manifolds and Carnot groups.

Relative heat content asymptotics for sub-Riemannian manifolds

Abstract

The relative heat content associated with a subset of a sub-Riemannian manifold, is defined as the total amount of heat contained in at time , with uniform initial condition on , allowing the heat to flow outside the domain. In this work, we obtain a fourth-order asymptotic expansion in square root of of the relative heat content associated with relatively compact non-characteristic domains. Compared to the classical heat content that we studied in [Rizzi, Rossi - J. Math. Pur. Appl., 2021], several difficulties emerge due to the absence of Dirichlet conditions at the boundary of the domain. To overcome this lack of information, we combine a rough asymptotic for the temperature function at the boundary, coupled with stochastic completeness of the heat semi-group. Our technique applies to any (possibly rank-varying) sub-Riemannian manifold that is globally doubling and satisfies a global weak Poincaré inequality, including in particular sub-Riemannian structures on compact manifolds and Carnot groups.

Paper Structure

This paper contains 34 sections, 27 theorems, 206 equations.

Table of Contents

  1. Introduction
  2. Strategy of the proof of Theorem \ref{['t:intro1']}
  3. The classical heat content.
  4. Failure of Duhamel's principle.
  5. First-order asymptotics.
  6. The outside contribution Icphi.
  7. Fourth-order asymptotics.
  8. From the heat kernel asymptotics to the relative heat content asymptotics
  9. Characteristic points
  10. Acknowledgments.
  11. Preliminaries
  12. Sub-Riemannian geometry
  13. The relative heat content
  14. Nilpotent approximation of M
  15. Sub-Riemannian flag.
  16. ...and 19 more sections

Key Result

Theorem 1.1

Let $M$ be a compact sub-Riemannian manifold, equipped with a smooth measure $\omega$, and let $\Omega\subset M$ be an open subset whose boundary is smooth and has no characteristic points. Then, as $t\to 0$, where $\sigma$ denotes the sub-Riemannian perimeter measure.

Theorems & Definitions (69)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 2.1
  • Definition 2.2: Relative heat content
  • Remark 2.3
  • Definition 2.4: Characteristic point
  • Definition 2.5: Privileged coordinates
  • ...and 59 more