Approximately Einstein ACH metrics, volume renormalization, and an invariant for contact manifolds

TL;DR

This paper constructs an approximately Einstein ACH metric on contact manifolds, showing that the volume expansion's log term coefficient is a contact invariant, extending concepts from CR geometry and exploring implications for CR Q-curvature.

Contribution

It introduces a new approximately Einstein ACH metric for contact manifolds and proves the invariance of the volume expansion's log term coefficient under contact structure changes.

Findings

The log term coefficient in volume expansion is a contact invariant.

Construction of an approximately Einstein ACH metric on contact manifolds.

Identification of a new obstruction form in the partially integrable setting.

Abstract

To any smooth compact manifold $M$ endowed with a contact structure $H$ and partially integrable almost CR structure $J$, we prove the existence and uniqueness, modulo high-order error terms and diffeomorphism action, of an approximately Einstein ACH (asymptotically complex hyperbolic) metric $g$ on $M\times (-1,0)$. We consider the asymptotic expansion, in powers of a special defining function, of the volume of $M\times (-1,0)$ with respect to $g$ and prove that the log term coefficient is independent of $J$ (and any choice of contact form $\theta$), i.e., is an invariant of the contact structure $H$. The approximately Einstein ACH metric $g$ is a generalisation of, and exhibits similar asymptotic boundary behaviour to, Fefferman's approximately Einstein complete K\"ahler metric $g_+$ on strictly pseudoconvex domains. The present work demonstrates that the CR-invariant log term coefficient in the asymptotic volume expansion of $g_+$ is in fact a contact invariant. We discuss some implications this may have for CR $Q$-curvature. The formal power series method of finding $g$ is obstructed at finite order. We show that part of this obstruction is given as a one-form on $H^*$. This is a new result peculiar to the partially integrable setting.