[Paper Review] CR-Invariants and the Scattering Operator for Complex Manifolds with Boundary

Peter D. Hislop, Peter Perry+1 more|ArXiv.org|Sep 7, 2007

Geometry and complex manifolds18 references3 citations

TL;DR

This paper establishes that CR-covariant differential operators and the CR $Q$-curvature on strictly pseudoconvex CR manifolds arise as residues of the scattering operator for the Laplacian on an ambient complex Kähler manifold with the CR manifold as its boundary at infinity. The key result is that the scattering operator encodes both the $P_k$ operators and the CR $Q$-curvature via finite part integrals and residue calculus, generalizing Graham and Zworski's results on Poincaré-Einstein manifolds to the CR setting.

ABSTRACT

The purpose of this paper is to describe certain CR-covariant differential operators on a strictly pseudoconvex CR manifold $M$ as residues of the scattering operator for the Laplacian on an ambient complex Kähler manifold $X$ having $M$ as a `CR-infinity.' We also characterize the CR $Q$-curvature in terms of the scattering operator. Our results parallel earlier results of Graham and Zworski \cite{GZ:2003}, who showed that if $X$ is an asymptotically hyperbolic manifold carrying a Poincaré-Einstein metric, the $Q$-curvature and certain conformally covariant differential operators on the `conformal infinity' $M$ of $X$ can be recovered from the scattering operator on $X$. The results in this paper were announced in \cite{HPT:2006}.

Motivation & Objective

• To characterize CR-covariant differential operators and the CR $Q$-curvature on strictly pseudoconvex CR manifolds using the scattering operator.
• To extend Graham and Zworski's scattering-theoretic construction of $Q$-curvature and conformally invariant operators to the CR geometry setting.
• To establish that the scattering operator on an ambient Kähler manifold with boundary $M$ encodes the CR $P_k$ operators and $Q$-curvature through residue and finite part integrals.
• To provide a geometric realization of CR-invariants via asymptotic analysis of the Laplacian on an ambient complex manifold with boundary at infinity.

Proposed method

• Construct an ambient complex Kähler manifold $X$ of dimension $m=n+1$ with strictly pseudoconvex CR boundary $M$ of dimension $2n+1$.
• Define a Kähler metric $g_\varphi$ on $X$ using a defining function $\varphi$ with $\varphi<0$ in the interior, and asymptotic expansion in $\varphi$ near $M$.
• Study the scattering operator associated with the Laplacian $\Delta_{g_\varphi}$ on $X$, analyzing its meromorphic continuation and residues.
• Use the asymptotic expansion of solutions $u_s$ to the eigenvalue problem $\Delta_{g_\varphi}u_s = s(m-s)u_s$ near the boundary to extract residues.
• Compute finite part integrals of $|du_s|^2 \omega^m$ and $u_s^2 \omega^m$ to isolate contributions from $x^{2m-2s}$ terms, where $x = -\varphi$.
• Relate the residue of the scattering operator to the CR $Q$-curvature via the coefficient of $x^{2m-2s}$ in the expansion of $u_s^2$, identifying it with $-m c_m \int_M Q^{CR}_\theta \psi$.

Experimental results

Research questions

• RQ1 How can CR-covariant differential operators $P_k$ on a strictly pseudoconvex CR manifold $M$ be recovered from the scattering operator of an ambient Kähler manifold $X$?
• RQ2 Can the CR $Q$-curvature be characterized as a residue of the scattering operator in the complex geometric setting?
• RQ3 To what extent does the scattering operator on $X$ encode the conformal invariants of the CR boundary $M$?
• RQ4 How do the asymptotics of solutions to the eigenvalue problem $\Delta_{g_\varphi}u_s = s(m-s)u_s$ yield geometric invariants on $M$?

Key findings

• The CR-covariant differential operators $P_k$ of order $2k$ arise as residues of the scattering operator for the Laplacian on the ambient Kähler manifold $X$.
• The CR $Q$-curvature $Q^{CR}_\theta$ is recovered as the residue of the scattering operator via the finite part of $\int_X u_s^2 \omega^m$, specifically as $-m c_m \int_M Q^{CR}_\theta \psi$.
• The scattering operator's residue at $s=m$ isolates the $x^{2m-2s}$ term in the expansion of $u_s^2$, which contributes $m/2 \cdot L$ to the integral, where $L$ is the asymptotic volume coefficient.
• The term $2x^{2m-2s}F_m(s)$ in the expansion contributes $-m c_m \int_M Q^{CR}_\theta \psi$, directly linking the scattering residue to the CR $Q$-curvature.
• The contribution from the $|du_s|^2$ term in the scattering integral is shown to vanish in the finite part limit as $s \to m$, ensuring the residue is determined solely by the $u_s^2$ term.
• The boundary terms involving $\partial_y u_s$ vanish in the limit $s \to m$ due to the $(m-s)$-order decay of derivatives, confirming the finiteness and correctness of the residue computation.