Abstract

 

Among those transversally elliptic operators initiated by Atiyah

and Singer, Kohn’s b operator on CR manifolds with S1 action

is a natural one of geometric significance for complex analysts.

Our first main result establishes an asymptotic expansion for the

heat kernel of such an operator with values in its Fourier coefficients,

which involves an unprecedented contribution in terms

of a distance function from the lower dimensional stratum of the

S1-action. Our second main result computes a local index density,

in terms of tangential characteristic forms, on such manifolds including

Sasakian manifolds of interest in String Theory, by showing

that the non-trivial contributions from the stratum in the heat

kernel expansion will eventually cancel out. As applications of

our CR index theorem we can produce many CR functions on

a weakly pseudoconvex CR manifold with transversal S1 action

and many CR sections on some class of CR manifolds. We will

also discuss some embedding Theorems in CR geometry.