Heat content asymptotics for sub-Riemannian manifolds

Abstract: 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 for small times, at arbitrary order. We compute explicitly the coefficients up to order k = 5, in terms of sub-Riemannian invariants of the domain. Furthermore, as an independent result, 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 Tyson and Wang in the Heisenberg group [Comm. PDE, 2018]. 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.

This is a joint work with T. Rossi (Institut Fourier & SISSA)

analysis of PDEsdifferential geometrymetric geometry

Audience: researchers in the topic

International sub-Riemannian seminar

Series comments: Description: Research seminars in sub-Riemannian geometry