Mathematics

We study the small-time asymptotics of the heat content of 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 the full asymptotic series. We compute explicitly the coefficients up to order five, in terms of sub-Riemannian invariants of the domain and its boundary. Furthermore, we prove that every coefficient can be obtained as the limit of the corresponding one for a suitable Riemannian extension. As a byproduct of our fifth-order analysis, we prove that the higher order coefficients in the expansion can blow-up in presence of characteristic points. This is a joint work with Luca Rizzi.