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.

32 pages

References in corpus (2)

• The Hessian of the distance from a surface in the Heisenberg group
• Tubular neighborhoods in the sub-Riemannian Heisenberg groups

Cited by in corpus (2)

• Inverse source problems for positive operators. I. Hypoelliptic diffusion and subdiffusion equations
• Heat content asymptotics for sub-Riemannian manifolds