论文标题
亚riemannian歧管的热含量渐近学
Heat content asymptotics for sub-Riemannian manifolds
论文作者
论文摘要
我们研究了配备有任意平滑度量的一般性级别次摩曼尼亚结构的平滑非特征域的热含量的小渐近含量。通过适应次摩nanian案例，由于SAVO引起的一种技术，我们确定了完整的渐近系列的存在：\ begin {equination}q_Ω（t）= \ sum_ {k = 0}^{\ infty} {\ infty} a_k t^{k/2}，\ qquad \ qquad \ qquad \ qquad \ qquad \ qquad \ qquad \ quald \ quart \ text {明确地，根据域及其边界的子侵入式不变式的订单$ k = 5 $的系数。此外，我们证明，对于合适的riemannian扩展，可以作为相应的系数获得每个系数。作为一种特殊情况，我们使用非稳态技术恢复了$ 2 $公式的订单，泰森和王在第一个海森伯格组中[J. J.泰森（J. Wang），通讯。 PDE，2018年]。我们五阶分析的有趣副产品是在存在特征点的情况下新现象的证据。特别是，我们证明了扩展中的高阶系数在存在下会爆炸。最后结果的关键工具是在第一个海森伯格组中具有孤立特征点的特定表面距离距离的确切公式，该表面具有独立的关注。
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: \begin{equation} Q_Ω(t) = \sum_{k=0}^{\infty} a_k t^{k/2}, \qquad \text{as } t\to 0. \end{equation} We compute explicitly the coefficients up to order $k=5$, in terms of sub-Riemannian invariants of the domain and its boundary. 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 due to Tyson and Wang in the first Heisenberg group [J. Tyson, J. Wang, Comm. PDE, 2018]. An intriguing byproduct 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 expansion can blow-up in their presence. A key tool for this last result is an exact formula for the sub-Riemannian distance from a specific surface with an isolated characteristic point in the first Heisenberg group, which is of independent interest.
