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Biquard, Olivier; Herzlich, Marc

A Burns-Epstein invariant for ACHE 4-manifolds. (English) Zbl 1074.53037

Duke Math. J. 126, No. 1, 53-100 (2005).
The authors study properties of a renormalized characteristic class for Einstein asymptotically complex hyperbolic manifolds of dimension \(4\) (ACHE \(4\)-manifolds). This class is defined by using a combination of the norms of various parts of the curvature.
Theorem 1.1: Let \((M,g)\) be an asymptotically complex hyperbolic Einstein manifold of dimension \(4\). Then \(\frac{1}{8\pi^2}\int_M(3| W^-| ^2-| W^+| ^2+\frac{1}{24}Scal^2)\) converges and provides an invariant of the ACHE metric, called the Burns-Epstein invariant of \(g\).
Theorem 1.2: There is a global invariant \(\nu (X)\) defined on any compact strictly pseudoconvex \(3\)-dimensional smooth CR manifold \(X\) such that whenever \((M,g)\) is an ACHE manifold whose boundary at infinity \(\partial_\infty M\) is \(X\) with its CR structure, then \(\frac{1}{8\pi^2}\int_M(3| W^-| ^2-| W^+| ^2+\frac{1}{24}Scal^2)=\chi(M)-3\tau(M)+ \nu(X)\).
Next the invariant \(\nu\) is related to the invariant \(\mu(J,J^\prime\)) of two CR structures \(J,J^\prime\) on the same contact distribution.
Theorem 1.3: For any CR structures \(J\) and \(J^\prime\) on \(X^3\) with the same underlying contact structure, one has \(\nu(J)-\nu(J^\prime)=3\mu(J,J^\prime)\).
Reviewer: Vasile Oproiu (Iaşi)

Cited in 1 Review
Cited in 17 Documents

MSC:

53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)

Keywords:

Einstein complex manifolds; chracteristic classes; pseudoconvex CR manifolds; global invariants
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References:

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