Heat content and horizontal mean curvature on the Heisenberg group
COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS(2018)
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 coecients 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.
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Key words
Brownian motion,heat content,Heisenberg group,horizontal mean curvature,horizontal perimeter
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