Relative Heat Content Asymptotic for Sub-Riemmanian Manifolds

For a domain in M, the relative heat content is defined as the total amount of heat contained in the domain at time t, allowing the heat to flow outside the domain. We study the small-time asymptotics of the relative heat content associated with smooth non-characteristic domains of a general rank-varying sub-Riemannian structure, equipped with an arbitrary smooth measure. By adapting to the sub-Riemannian case a technique due to Savo, we establish the existence of an asymptotic series, up to order 4. Significant difficulties emerges, as the boundary behavior of the temperature function is not known: we use an “asymptotic” symmetry argument of the heat diffusion to obtain information on the small-time behavior of temperature at the boundary of the domain. This is a joint work with Andrei Agrachev and Luca Rizzi.