Heat content and horizontal mean curvature on the Heisenberg group

Abstract

We identify the short time asymptotics of the sub-Riemannian heat content for a smoothly bounded domain in the first Heisenberg group. Our asymptotic formula generalizes prior work by van den Berg-Le Gall and van den Berg-Gilkey to the sub-Riemannian context, and identifies the first few coefficients in the sub-Riemannian heat content in terms of the horizontal perimeter and the total horizontal mean curvature of the boundary. The proof is probabilistic, and relies on a characterization of the heat content in terms of Brownian motion.

Heat content asymptotics for sub-Riemannian manifolds

“…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 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 Tyson and Wang in the Heisenberg group [TW18]. A consequence of our fifth-order analysis is the evidence for new phenomena in presence of characteristic points.…”

“…As a particular case we recover, using non-probabilistic techniques, the order 2 formula recently obtained by Tyson and Wang in the Heisenberg group [TW18]. A consequence of our fifth-order analysis is the evidence for new phenomena in presence of characteristic points.…”

“…[DGN12]. As a consequence of all these facts, all terms appearing in the order 2 formula for H in [TW18] are well-defined also for characteristic domains. This seems to suggest that the same small-time asymptotic formula might hold also for characteristic domains.…”

“…The study of the small-time heat content asymptotics in the sub-Riemannian setting was initiated recently by Tyson and Wang, in [TW18], where they studied the first Heisenberg group H. There, they established the existence of a small-time asymptotic series up to order 2 in √ t, for non-characteristic domains. The approach in [TW18] is probabilistic, based on the interpretation of the solution of the Dirichlet problem in terms of the exit time of the corresponding Markov process.…”

Integrability of the sub-Riemannian mean curvature at degenerate characteristic points in the Heisenberg group

“…The horizontal mean curvature H : Σ → R is a geometrical invariant which arises naturally in different areas of geometric analysis. It appears in the theory of minimal surfaces [DGN07,HP08,Mon15,Pau04,CHMY05], in the study of the heat content asymptotics in sub-Riemannian manifolds [TW18,RR20], and in Steiner-type formulas for the volume of tubes around hypersurfaces [BFF + 15].…”