Noncommutative residue for Heisenberg manifolds. Applications in CR and contact geometry

Abstract

This paper has four main parts. In the first part, we construct a noncommutative residue for the hypoelliptic calculus on Heisenberg manifolds, that is, for the class of Ψ H DO operators introduced by Beals–Greiner and Taylor. This noncommutative residue appears as the residual trace on integer order Ψ H DOs induced by the analytic extension of the usual trace to non-integer order Ψ H DOs. Moreover, it agrees with the integral of the density defined by the logarithmic singularity of the Schwartz kernel of the corresponding Ψ H DO. In addition, we show that this noncommutative residue provides us with the unique trace up to constant multiple on the algebra of integer order Ψ H DOs. In the second part, we give some analytic applications of this construction concerning zeta functions of hypoelliptic operators, logarithmic metric estimates for Green kernels of hypoelliptic operators, and the extension of the Dixmier trace to the whole algebra of integer order Ψ H DOs. In the third part, we present examples of computations of noncommutative residues of some powers of the horizontal sublaplacian and the contact Laplacian on contact manifolds. In the fourth part, we present two applications in CR geometry. First, we give some examples of geometric computations of noncommutative residues of some powers of the horizontal sublaplacian and of the Kohn Laplacian. Second, we make use of the framework of noncommutative geometry and of our noncommutative residue to define lower-dimensional volumes in pseudohermitian geometry, e.g., we can give sense to the area of any 3-dimensional CR manifold endowed with a pseudohermitian structure. On the way we obtain a spectral interpretation of the Einstein–Hilbert action in pseudohermitian geometry.