Barilari D., Boscain U., Sigalotti M.(Eds) Geometry, Analysis and Dynamics on sub-Riemannian Manifolds. Volume I

London, UK, European Mathematical Society, 2016. — 334 p. — (EMS Series of Lectures in Mathematics 24). — ISBN: 3037191627.A publication of the European Mathematical Society Sub-Riemannian manifolds model media with constrained dynamics: motion at any point is allowed only along a limited set of directions, which are prescribed by the physical problem. From the theoretical point of view, sub-Riemannian geometry is the geometry underlying the theory of hypoelliptic operators and degenerate diffusions on manifolds. In the last twenty years, sub-Riemannian geometry has emerged as an independent research domain, with extremely rich motivations and ramifications in several parts of pure and applied mathematics, such as geometric analysis, geometric measure theory, stochastic calculus and evolution equations together with applications in mechanics, optimal control and biology. The aim of the lectures collected here is to present sub-Riemannian structures for the use of both researchers and graduate students. A publication of the European Mathematical Society (EMS).Some topics of geometric measure theory in Carnot groupsAn introduction to Carnot groups
Differential calculus on Carnot groups
Differential calculus within Carnot groups
Sets of finite perimeter and minimal surfaces in Carnot groups
Hypoelliptic operators and some aspects of analysis and geometry of sub-Riemannian spaces
Sub-Riemannian geometry and hypoelliptic operators
Carnot groups
Fundamental solutions and the Yamabe equation
Carnot–Carathéodory distance
Sobolev and BV spaces
Fractional integration in spaces of homogeneous type
Fundamental solutions of hypoelliptic operators
The geometric Sobolev embedding and the isoperimetric inequality
The Li–Yau inequality for complete manifolds with Ricci > 0
Heat semigroup approach to the Li–Yau inequality
A heat equation approach to the volume doubling property
A sub-Riemannian curvature-dimension inequality
Sub-Laplacians and hypoelliptic operators on totally geodesic Riemannian foliationsRiemannian foliations and their Laplacians
Horizontal Laplacians and heat kernels on model spaces
Transverse Weitzenböck formulas
The horizontal heat semigroup
The horizontal Bonnet–Myers theorem
Riemannian foliations and hypocoercivity