SubLaplacians

The subLaplacian is a differential operator that arises in the study of subRiemannian geometry, a generalization of Riemannian geometry where the metric is not necessarily defined on the entire tangent bundle. Unlike the standard Laplacian, which is built from the metric tensor and its derivatives, the subLaplacian is constructed using a system of vector fields that span the tangent space only in a restricted sense.

In essence, the subLaplacian is defined as the sum of squares of these vector fields, acting on a function. This definition is local and depends on the choice of the vector fields. The operators are degenerate elliptic, meaning their principal symbol has a non-trivial null space. This degeneracy leads to different analytical properties compared to the standard Laplacian, such as a different heat kernel behavior and unique geodesic structures.

SubLaplacians are fundamental to understanding the geometry and analysis on subRiemannian manifolds. They play a role in areas like geometric control theory, harmonic analysis, and the study of partial differential equations with degenerate coefficients. The study of subLaplacians involves developing new techniques to handle the lack of a full-rank metric and the resulting anisotropic nature of the geometry.