Compactness of pseudohermitian structures with integral bounds on curvature

In this paper, we show a compactness criterion for contact forms in a fixed CR structure(i.e., conformal pseudohermitian structures), assuming a volume bound and L^p bounds, p>2, on the Tanaka-Webster curvature, the pseudohermitian torsion and its covariant derivative. We also need the L² bound on the derivative of the Tanaka-Webster curvature along the characteristic vector field. As an application, we can show that the CR automorphism group is compact if M is not CR spherical or the CR Yamabe constant is negative.