Heat content asymptotics in sub-Riemannian geometry

Sub-Riemannian geometry is a particularly rich class of metric structures, which generalizes Riemannian geometry, where a smoothly varying metric is defined only on a subset of preferred directions of the tangent space at each point of a smooth manifold M (called horizontal directions). Under the so-called Hörmander condition, M is horizontally-path connected, and the usual length-minimization procedure yields a well-defined metric. The Laplace-Beltrami operator is generalized by the sub-Laplacian which is sub-elliptic, but has nonetheless suitable regularity properties (in particular, it is hypoelliptic). In this thesis, we investigate the heat content asymptotics and related topics in sub-Riemannian geometry. For a domain in M, the heat content is defined as the total amount of heat contained in the domain at time t, assuming that its boundary is kept at zero temperature for all times. In the Heisenberg group, which is the simplest example of sub-Riemannian structure, Tyson and Wang studied the small-time behavior of this quantity, proving the existence of an asymptotic expansion up to order 2 in square root of t. Here, using a different approach, we show the existence of a complete asymptotic expansion of the heat content in any sub-Riemannian manifold, also providing an algorithm for computing its coefficients at any order. As in the Riemannian case, the small-time behavior of the heat content contains geometrical information of the boundary of the domain. A crucial assumption on the domain for developing a complete asymptotic expansion of the heat content is the absence of characteristic points. Roughly speaking, a characteristic point is a point of the boundary of the domain (which is assumed to be a smooth submanifold of M) where the distance function from the boundary itself loses regularity. We show that, if characteristic points are present a new phenomenon occurs in the asymptotic expansion of the heat content. In particular, the latter can no longer be true to order equal to or greater than 5, in general. In addition, we study the horizontal mean curvature of the boundary when characteristic points are present. In the particular case of the Heisenberg group, we introduce the notion of a mildly degenerate characteristic point, proving new integrability results for the horizontal mean curvature of surfaces. This result, in the case of analytic surfaces in the Heisenberg group, answers affirmatively to a conjecture formulated by Danielli, Garofalo and Nhieu. Finally, we study a related, yet different, quantity called the relative heat content. For a domain in M, the relative heat content is defined as the total amount of heat contained in the domain at time t, allowing the heat to flow outside the domain. Significant difficulties emerges, as the boundary behavior of the temperature function is no longer known, as opposed to the case of the classical heat content. We use an “asymptotic” symmetry argument of the heat diffusion to obtain information on the small-time behavior of temperature at the boundary of the domain and we obtain a fourth-order asymptotic expansion in square root of t.