[Paper Review] Q and Q-prime curvature in CR geometry

Kengo Hirachi|arXiv (Cornell University)|May 9, 2014

Holomorphic and Operator Theory18 references3 citations

TL;DR

This paper introduces Q-prime curvature as a secondary Q-curvature invariant in CR geometry, showing that its total integral, the total Q-prime curvature, achieves a local maximum at the standard CR sphere under CR structure deformations. Using the deformation complex of CR structures and representation theory, the authors derive a variational formula and prove that the Hessian of the total Q-prime curvature is semidefinite, with kernel tied to obstruction functions and Chern-Moser tensor vanishing, confirming the sphere as a critical point.

ABSTRACT

The Q-curvature has been playing a central role in conformal geometry since its discovery by T. Branson. It has natural analogy in CR geometry, however, the CR Q-curvature vanishes on the boundary of a strictly pseudoconvex domain in C^{n+1} with a natural choice of contact form. This fact enables us to define a "secondary" Q-curvature, which we call Q-prime curvature (it was first introduced by J. Case and P. Yang in the case n=1). The integral of the Q-prime curvature, the total Q-prime curvature, is a CR invariant of the boundary. When n=1, it agrees with the Burns-Epstein invariant, which is a Chern-Simons type invariant in CR geometry. For all n>=1, it has non-trivial variation under the deformation of domains. Combining the variational formula with the deformation complex of CR structures, we show that the total Q-prime curvature takes local maximum at the standard CR sphere in a formal sense.

Motivation & Objective

• To define a secondary Q-curvature, Q-prime curvature, in higher-dimensional CR manifolds where the standard Q-curvature vanishes on strictly pseudoconvex boundaries.
• To establish the total Q-prime curvature as a global CR invariant that generalizes the Burns-Epstein invariant in dimension 3.
• To analyze the variation of the total Q-prime curvature under deformations of CR structures using the deformation complex.
• To show that the total Q-prime curvature achieves a local maximum at the standard CR sphere in a formal sense, using Hessian analysis and representation theory.
• To explore the connection between the Hessian of the total Q-prime curvature and the integrability conditions of partially integrable CR structures.

Proposed method

• Define Q-prime curvature as a secondary invariant in CR geometry, analogous to the secondary Q-curvature in conformal geometry.
• Use the ambient metric construction to express local invariants of the CR boundary in terms of curvature jets.
• Apply the deformation complex of CR structures, realized as a generalized Bernstein-Gelfand-Gelfand complex, to analyze the second variation of the total Q-prime curvature.
• Compute the Hessian of the total Q-prime curvature as a CR-invariant, self-adjoint differential operator between bundles in the deformation complex.
• Use representation theory to determine the kernel of the Hessian operator and establish its semidefiniteness.
• Relate the vanishing of the Hessian's kernel to the vanishing of obstruction functions and the Chern-Moser tensor, characterizing sphericality to first order.

Experimental results

Research questions

• RQ1 Does a secondary Q-curvature exist in higher-dimensional CR manifolds where the standard Q-curvature vanishes?
• RQ2 Can the total Q-prime curvature be shown to be a CR invariant that generalizes the Burns-Epstein invariant in dimension 3?
• RQ3 Is the total Q-prime curvature maximized at the standard CR sphere under CR structure deformations?
• RQ4 What is the geometric meaning of the kernel of the Hessian of the total Q-prime curvature?
• RQ5 How does the Hessian of the total Q-prime curvature relate to the integrability and obstruction theory of CR structures?

Key findings

• The total Q-prime curvature takes a local maximum at the standard CR sphere in a formal sense, as shown by the semidefiniteness of the Hessian of the functional.
• The Hessian of the total Q-prime curvature is a CR-invariant, self-adjoint differential operator between bundles in the deformation complex, and its kernel corresponds to the sum of the kernels of $ D_0^+ $ and $ D_0^- $.
• For $ n eq 1 $, the condition $ ext{ker}(D_0^+) + ext{ker}(D_0^-) = ext{ker}(R_1^- D_0^+) $ implies that deformations in the kernel correspond to those that are spherical to first order.
• The Hessian of the total Q-prime curvature is semidefinite, and its kernel captures the directions of non-trivial deformation that preserve sphericality to first order.
• The result confirms that the standard CR sphere is a critical point of the total Q-prime curvature functional, consistent with known minimality results for the Burns-Epstein invariant in dimension 3.
• For partially integrable CR structures, the total Q-curvature is non-trivial, and its Hessian reveals a connection to the integrability condition, though it does not characterize integrable structures.