On the spectrum of a strictly pseudoconvex CR manifold (Q1378231)

Let \((M,T^{1,0}M)\) be a compact, strictly pseudoconvex CR-manifold and let \(\theta\) be a choice of contact form. This defines a differential operator: \(\overline\partial_b f= df |_{T^{0,1} M}\), a vector field \(T\), transverse to \(\ker\theta\): \(\theta (T)=0\) \(T\rfloor d \theta =0\) and a pseudohermitian connection, \(\nabla_\theta\). Let Ric be the Ricci curvature and \(A\) the torsion tensor of \(\nabla_\theta\). The authors prove the following theorem: Theorem: Assume that the problem \[ \begin{cases} \Delta_bv= \lambda_kv, \quad & T(v)=0,\\ \sup v=1 \\ \inf v=-c, \quad & 0<c\leq 1\end{cases} \] has a \(C^\infty\) solution. If \[ \text{Ric} (X-iJX,X+ iJX) +2(n-2) A(X,JX)\geq 0 \] for all \(X\in \ker\theta\) then \(\lambda_k\geq {\pi^2 \over d^2_\theta}\). Here \(\Delta_b f=2(\overline \partial^*_b \overline \partial_bf) -inT(f)\) und \(d_\theta\) is the diameter of \(M\) with respect to the Webster metric.