C-R Immersions and Sub-Riemannian Geometry

On any strictly pseudoconvex CR manifold M, of CR dimension n, equipped with a positively oriented contact form θ, we consider natural ϵ-contractions, i.e., contractions gϵM of the Levi form Gθ, such that the norm of the Reeb vector field T of (M, θ) is of order O(ϵ−1). We study isopseudohermitian (i.e., f∗Θ=θ) Cauchy–Riemann immersions f:M→(A,Θ) between strictly pseudoconvex CR manifolds M and A, where Θ is a contact form on A. For every contraction gϵA of the Levi form GΘ, we write the embedding equations for the immersion f:M→A,gϵA. A pseudohermitan version of the Gauss equation for an isopseudohermitian C-R immersion is obtained by an elementary asymptotic analysis as ϵ→0+. For every isopseudohermitian immersion f:M→S2N+1 into a sphere S2N+1⊂CN+1, we show that Webster’s pseudohermitian scalar curvature R of (M, θ) satisfies the inequality R≤2n(f∗gΘ)(T,T)+n+1+12{∥H(f)∥gΘf2+∥traceGθΠH(M)∇⊤−∇∥f∗gΘ2} with equality if and only if B(f)=0 and ∇⊤=∇ on H(M)⊗H(M). This gives a pseudohermitian analog to a classical result by S-S. Chern on minimal isometric immersions into space forms.