Recent zbMATH articles in MSC 58J60https://zbmath.org/atom/cc/58J602021-05-28T16:06:00+00:00WerkzeugHeat content asymptotics for sub-Riemannian manifolds.https://zbmath.org/1459.353642021-05-28T16:06:00+00:00"Rizzi, Luca"https://zbmath.org/authors/?q=ai:rizzi.luca"Rossi, Tommaso"https://zbmath.org/authors/?q=ai:rossi.tommasoSummary: 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:
\[
Q_{\Omega}(t)=\sum_{k=0}^\infty a_k t^{k/2},\text{ as }t\to 0.
\]
We compute explicitly the coefficients up to order \(k=5\), in terms of sub-Riemannian invariants of the domain. Furthermore, we prove that every coefficient can be obtained as the limit of the corresponding one for a suitable Riemannian extension. As a particular case we recover, using non-probabilistic techniques, the order 2 formula recently obtained by \textit{J. Tyson} and \textit{J. Wang} [Commun. Partial Differ. Equations 43, No. 3, 467--505 (2018; Zbl 1391.53043)] in the Heisenberg group. A consequence of our fifth-order analysis is the evidence for new phenomena in presence of characteristic points. In particular, we prove that the higher order coefficients in the asymptotics can blow-up in their presence. A key tool for this last result is an exact formula for the distance from a specific surface with an isolated characteristic point in the Heisenberg group, which is of independent interest.