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

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 $$\nu(J)-\nu(J^\prime)=3\mu(J,J^\prime)$$.

### MSC:

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

### References:

 [1] M. T. Anderson, $$L^ 2$$ curvature and volume renormalization of AHE metrics on 4-manifolds , Math. Res. Lett. 8 (2001), 171–188. · Zbl 0999.53034 [2] M. F. Atiyah, V. K. Patodi, and I. M. Singer, Spectral asymmetry and Riemannian geometry, III , Math. Proc. Cambridge Philos. Soc. 79 (1976), 71–99. · Zbl 0325.58015 [3] A. L. Besse, Einstein Manifolds , Ergeb. Math. Grenzgeb. (3) 10 , Springer, Berlin, 1987. · Zbl 0613.53001 [4] O. Biquard, Métriques d’Einstein asymptotiquement symétriques , Astérisque 265 , Soc. Math. France, Montrouge, 2000. · Zbl 0967.53030 [5] ——–, Autodual Einstein versus Kähler-Einstein , · Zbl 1082.53026 [6] J. S. Bland, Contact geometry and CR structures on $$S^ 3$$ , Acta Math. 172 (1994), 1–49. · Zbl 0814.32002 [7] J.-P. Bourguignon and H. B. Lawson Jr., Stability and isolation phenomena for Yang-Mills fields , Comm. Math. Phys. 79 (1981), 189–230. · Zbl 0475.53060 [8] D. M. Burns Jr. and C. L. Epstein, A global invariant for three-dimensional CR-manifolds , Invent. Math. 92 (1988), 333–348. · Zbl 0643.32006 [9] –. –. –. –., Characteristic numbers of bounded domains , Acta Math. 164 (1990), 29–71. · Zbl 0704.32005 [10] D. M. J. Calderbank and M. A. Singer, Einstein metrics and complex singularities , Invent. Math. 156 (2004), 405–443. · Zbl 1061.53026 [11] D. Catlin, “Extension of CR structures” in Several Complex Variables and Complex Geometry, Part 3 (Santa Cruz, Calif., 1989) , Proc. Sympos. Pure Math. 52 , Amer. Math. Soc., Providence, 1991, 27–34. · Zbl 0790.32020 [12] J. H. Chêng and J. M. Lee, The Burns-Epstein invariant and deformation of CR structures , Duke Math. J. 60 (1990), 221–254. · Zbl 0704.53028 [13] S. Y. Cheng and S. T. Yau, On the existence of a complete Kähler metric on noncompact complex manifolds and the regularity of Fefferman’s equation , Comm. Pure Appl. Math. 33 (1980), 507–544. · Zbl 0506.53031 [14] S. S. Chern and J. K. Moser, Real hypersurfaces in complex manifolds , Acta Math. 133 (1974), 219–271.; Erratum , Acta Math. 150 (1983), 297. ; Mathematical Reviews (MathSciNet): · Zbl 0302.32015 [15] E. Delay and M. Herzlich, Ricci curvature in the neighborhood of rank-one symmetric spaces , J. Geom. Anal. 11 (2001), 573–588. · Zbl 1035.53051 [16] C. L. Epstein, CR-structures on three-dimensional circle bundles , Invent. Math. 109 (1992), 351–403. · Zbl 0786.32013 [17] C. L. Epstein, R. B. Melrose, and G. A. Mendoza, Resolvant of the Laplacian on strictly pseudoconvex domains , Acta Math. 167 (1991), 1–106. · Zbl 0758.32010 [18] C. L. Fefferman, Monge-Ampère equations, the Bergman kernel, and geometry of pseudoconvex domains , Ann. of Math. (2) 103 (1976), 395–416.; Correction , Ann. of Math. (2) 104 (1976), 393–394. ; Mathematical Reviews (MathSciNet): · Zbl 0322.32012 [19] C. Fefferman and C. R. Graham, “Conformal invariants” in Élie Cartan et les mathématiques d’aujourd’hui (Lyon, 1984) , Astérisque 1985 , numéro hors série, Soc. Math. France, Montrouge, 1985, 95–116. [20] P. Gauduchon, Variétés riemanniennes autoduales (d’après C. H. Taubes et al.) , Astérisque 216 (1993), 151–186., Séminaire Bourbaki 1992/93, no. 767. · Zbl 0789.53026 [21] M. Herzlich, A remark on renormalized volume and Euler characteristic for ACHE $$4$$-manifolds , · Zbl 1129.53027 [22] N. J. Hitchin, Einstein metrics and the eta-invariant , Boll. Un. Mat. Ital. B (7) 11 (1997), suppl., 95–105. · Zbl 0973.53519 [23] M. Kuranishi, “PDEs associated to the CR embedding theorem” in Analysis and Geometry in Several Complex Variables (Katata, Japan, 1997) , Trends Math., Birkhäuser, Boston, 1999, 129–157. · Zbl 0973.32019 [24] J. M. Lee and R. Melrose, Boundary behaviour of the complex Monge-Ampère equation , Acta Math. 148 (1982), 159–192. · Zbl 0496.35042 [25] L. Lempert, On three-dimensional Cauchy-Riemann manifolds , J. Amer. Math. Soc. 5 (1992), 923–969. JSTOR: · Zbl 0781.32014 [26] Y. Rollin, Rigidité d’Einstein du plan hyperbolique complexe , C. R. Math. Acad. Sci. Paris 334 (2002), 671–676. · Zbl 1072.53523 [27] –. –. –. –., Rigidité d’Einstein du plan hyperbolique complexe , J. Reine Angew. Math. 567 (2004), 175–213. · Zbl 1038.53044
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.