×

Lower-dimensional Fefferman measures via the Bergman kernel. (English) Zbl 1372.32041

Berhanu, Shiferaw (ed.) et al., Analysis and geometry in several complex variables. Workshop on analysis and geometry in several complex variables, Texas A&M University at Qatar, Doha, Qatar, January 4–8, 2015. Proceedings. Providence, RI: American Mathematical Society (AMS) (ISBN 978-1-4704-2255-4/pbk; 978-1-4704-3663-6/ebook). Contemporary Mathematics 681, 137-151 (2017).
The author gives a new construction of the Fefferman measure with aid of the Bergman kernel. Motivated by the Hausdorff measure theory, the author defines the Hausdorff-Fefferman measure (essentially) and Hausdorff-Fefferman dimension in the following way.
Definition 1. Let \(\Omega\subset\mathbb{C}^d\) be a bounded domain, \(K_{\Omega}\) be its Bergman kernel, \(w_{\mathbb{C}^d}\) be the volume form in \(\mathbb{C}^d\). For any \(M>0\), set \(\Omega_M:=\{z\in\Omega:K(z,z)>M\}\). Then the Hausdorff-Fefferman measure on \(\partial\Omega\) is defined as \[ \tilde \sigma_{\Omega}(A):=\text{weak-}*\text{ limit of }\frac{1}{\text{vol}(\Omega_M)}\chi_{\Omega_M}w_{\mathbb{C}^d}\qquad\text{as }M\to\infty, \] when it exists, where \(\chi_A\) denotes the indicator function of \(A\).
Definition 2. The Hausdorff-Fefferman dimension of a bounded domain \(\Omega\subset\mathbb{C}^d\) is said to exist if \[ \sup\Big\{\alpha>0:\liminf_{M\to\infty}M^{\frac{1}{\alpha}}\text{vol}(\Omega_M)=\infty\Big\}=\inf\Big\{\alpha>0:\limsup_{M\to\infty}M^{\frac{1}{\alpha}}\text{vol}(\Omega_M)=0\Big\}. \] In this case, the above quantity is the Hausdorff-Fefferman dimension, denoted by \(\dim_{HF}(\Omega)\).
With the help of some well-known formulas and estimates for the Bergman kernel, the author is able to
(1)
show that the Hausdorff-Fefferman measure \(\tilde \sigma_{\Omega}\) defined above is comparable to the original Fefferman measure \(\sigma_{\Omega}\), when \(\Omega\) is a strongly pseudoconvex domain in \(\mathbb{C}^d\);
(2)
compute the Hausdorff-Fefferman measure \(\tilde \sigma_{\Omega}\) explicitly, when \(\Omega\) is a product of balls.

Furthermore, by using some measure theory arguments, the author shows that similarly to the original Fefferman measure \(\sigma_{\Omega}\), the Hausdorff-Fefferman measure \(\sigma_{\Omega}\) also enjoys a transformation law under biholomorphic mappings.
Let \(F:\Omega^1\to\Omega^2\) be a biholomorphism between two \(\mathbb{C}^d\) domains, such that \(F\in C^1(\overline{\Omega^1})\) and \(J_{\mathbb{C}}F\) is non-vanishing. Suppose
(1)
\(\alpha:=\dim_{HF}(\Omega^1)<\infty\);
(2)
for \(a>0\), \(\text{vol}(\Omega_M^1)/\text{vol}(\Omega_{aM}^1)\) has a limit in \([0,\infty]\) as \(M\to\infty\);
(3)
\(\text{vol}(\Omega_M^1)/\text{vol}(\Omega_M^2)\) has a limit in \([0,\infty]\) as \(M\to\infty\).
Then, \[ F^*\tilde\sigma_{\Omega^2}\approx|\det J_{\mathbb{C}}F|^{2(1-1/a)}\tilde\sigma_{\Omega^1}, \] where \(\approx\) denotes equality up to renormalizations as probability measures.
For the entire collection see [Zbl 1358.32003].

MSC:

32T15 Strongly pseudoconvex domains
28A78 Hausdorff and packing measures
32A25 Integral representations; canonical kernels (Szegő, Bergman, etc.)
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] Barrett, David E., A floating body approach to Fefferman’s hypersurface measure, Math. Scand., 98, 1, 69-80 (2006) · Zbl 1145.32015 · doi:10.7146/math.scand.a-14984
[2] David E. Barrett, Holomorphic projection and duality for domains in complex projective space, Trans. Amer. Math. Soc. (2015). · Zbl 1338.32003
[3] Hammond, Christopher, Variational problems for Fefferman hypersurface measure and volume-preserving CR invariants, J. Geom. Anal., 21, 2, 372-408 (2011) · Zbl 1219.32022 · doi:10.1007/s12220-010-9151-2
[4] Barrett, David; Lee, Lina, On the Szeg\H o metric, J. Geom. Anal., 24, 1, 104-117 (2014) · Zbl 1303.32006 · doi:10.1007/s12220-012-9329-x
[5] W. Blaschke, Vorlesungen uber differentialgeometrie ii: Affine differentialgeometrie, Springer, Berlin, 1923. · JFM 49.0499.01
[6] Diederich, Klas, Some recent developments in the theory of the Bergman kernel function: a survey. Several complex variables, Proc. Sympos. Pure Math., Vol. XXX, Part 1, Williams Coll., Williamstown, Mass., 1975, 127-137 (1977), Amer. Math. Soc., Providence, R. I. · Zbl 0352.32008
[7] Fefferman, Charles, Parabolic invariant theory in complex analysis, Adv. in Math., 31, 2, 131-262 (1979) · Zbl 0444.32013 · doi:10.1016/0001-8708(79)90025-2
[8] Handbook of convex geometry. Vol. A, B, Vol.A: lxvi+735 pp.; Vol.B: pp.i-lxvi and 737-1438 (1993), North-Holland Publishing Co., Amsterdam · Zbl 0777.52002
[9] P. Gupta, Volume approximations of strictly pseudoconvex domains, J. Geom. Anal. (2016), 1-36. DOI: 10.1007/s12220-016-9709-8.
[10] Hammond, Christopher, Variational problems for Fefferman hypersurface measure and volume-preserving CR invariants, J. Geom. Anal., 21, 2, 372-408 (2011) · Zbl 1219.32022 · doi:10.1007/s12220-010-9151-2
[11] Hirachi, Kengo, Transformation law for the Szeg\H o projectors on CR manifolds, Osaka J. Math., 27, 2, 301-308 (1990) · Zbl 0713.32010
[12] H{\`“o}rmander, Lars, \(L^2\) estimates and existence theorems for the \(\bar \partial \)operator, Acta Math., 113, 89-152 (1965) · Zbl 0158.11002
[13] Leichtwei{\ss }, Kurt, Affine geometry of convex bodies, x+310 pp. (1998), Johann Ambrosius Barth Verlag, Heidelberg · Zbl 0899.52005
[14] Ludwig, Monika, Asymptotic approximation of smooth convex bodies by general polytopes, Mathematika, 46, 1, 103-125 (1999) · Zbl 0992.52002 · doi:10.1112/S0025579300007609
[15] Sch{\`“u}tt, Carsten; Werner, Elisabeth, The convex floating body, Math. Scand., 66, 2, 275-290 (1990) · Zbl 0739.52008 · doi:10.7146/math.scand.a-12311
[16] van der Vaart, A. W., Asymptotic statistics, Cambridge Series in Statistical and Probabilistic Mathematics 3, xvi+443 pp. (1998), Cambridge University Press, Cambridge · Zbl 0943.62002 · doi:10.1017/CBO9780511802256
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.