## Nagoya Mathematical Journal

### Localization lemmas for the Bergman metric at plurisubharmonic peak points

Gregor Herbort

#### Abstract

Let $D$ be a bounded pseudoconvex domain in $\mathbb{C}^n$ and $\zeta \in \mathrm {D}_{\!\!\!.}$. By $K_D$ and $B_D$ we denote the Bergman kernel and metric of $D$, respectively. Given a ball $B=B(\zeta, R)$, we study the behavior of the ratio $K_D/K_{D\cap B} (w)$ when $w\in D\cap B$ tends towards $\zeta$. It is well-known, that it remains bounded from above and below by a positive constant. We show, that the ratio tends to $1$, as $w$ tends to $\zeta$, under an additional assumption on the pluricomplex Green function $\mathcal{G}_D(\cdot , w)$ of $D$ with pole at $w$, the pluricomplex Green function $\mathcal{G}_D(\cdot , w)$ of $D$ with pole at $w$, namely that the diameter of the sublevel sets $A_w := \{z\in D \,\,|\,\, \mathcal{G}_D(z,w)<-1\}$ tends to zero, as $w\rightarrow \zeta$. A similar result is obtained also for the Bergman metric. In this case we also show that the extremal function associated to the Bergman kernel has the concentration of mass property introduced in {DiOh1}, where the question was discussed how to recognize a weight function from the associated Bergman space. The hypothesis concerning the set $A_w$ is satisfied for example, if the domain is regular in the sense of Diederich-Fornæss, ([DiFo2]).

#### Article information

Source
Nagoya Math. J., Volume 171 (2003), 107-125.

Dates
First available in Project Euclid: 27 April 2005

https://projecteuclid.org/euclid.nmj/1114631912

Mathematical Reviews number (MathSciNet)
MR2002015

Zentralblatt MATH identifier
1045.32009

#### Citation

Herbort, Gregor. Localization lemmas for the Bergman metric at plurisubharmonic peak points. Nagoya Math. J. 171 (2003), 107--125. https://projecteuclid.org/euclid.nmj/1114631912

#### References

• E. Bedford and J.E. Fornæss, Biholomorphic maps of weakly pseudoconvex domains , Duke Math. J., 45 (1978), 711–719.
• Z. Błocki, Estimates for the complex Monge-Ampère operator , Bull. Pol. Acad. Sci., 41 (1993), 151–157.
• Z. Błocki and P. Pflug, Hyperconvexity and Bergman completeness , Nagoya Math. J., 151 (1998), 221–225.
• M. Carlehed, U. Cegrell and F. Wikström, Jensen measures, Hyperconvexity and Boundary Behavior of the Pluricomplex Green's Function , Ann. Pol. Math., 71 (1999), 87–103.
• K. Diederich, Randverhalten der Bergmanschen Kernfunktion und Metrik in streng pseudokonvexen Gebieten , Math. Ann., 198 (1970), 1–36.
• K. Diederich and J.E. Fornæss, Pseudoconvex domains: Bounded strictly plurisubharmonic exhaustion functions , Inv. Math., 39 (1977), 129–141.
• ––––, Pseudoconvex domains: Existence of Stein neighborhoods , Duke Math. J., 44 (1977), 641–662.
• K. Diederich, J.E. Fornæss and G. Herbort, Boundary behavior of the Bergman metric , Proc. Pure Math., 41 (1984), 59–67.
• K. Diederich and G. Herbort, Pseudoconvex domains of semiregular type , Contributions to Complex Analysis and Analytic Geometry (Skoda,H.-Trépreau, Eds.), Aspects of Mathematics, E26 (1994), 127–162.
• ––––, Quantitative estimates for the Green function and an application to the Bergman metric , Ann. Inst. Fourier (Grenoble), 50 (2000), 1205–1228.
• ––––, An alternative proof of a theorem of T. Ohsawa , Michigan Mathematics J., 46 (1999), 347–360.
• K. Diederich and T. Ohsawa, Moment problems for weighted Bergman kernels , Complex analysis and geometry (Paris, 1997), Progress in Mathematics, 188 (2000), 111–122.
• ––––, An estimate for the Bergman distance on pseudoconvex domains , Ann. of Math., 141 (1995), 181–190.
• J.-P. Demailly, Mesures de Monge-Ampère et mesures pluriharmoniques , Math. Z., 194 (1987), 519–564.
• J.E. Fornæss and N. Sibony, Construction of p.s.h. functions on weakly pseudoconvex domains , Duke Math. J., 58 (1989), 633–656.
• G. Herbort, The Bergman metric on hyperconvex domains , Math. Z., 232 (1999), 183–196.
• ––––, Boundary behavior of the pluricomplex Green function on pseudoconvex domains with a smooth boundary , Internatl. J. Math., 11 (2000), 509–522.
• L. Hörmander, $L^2$-estimates and existence theorems for the $\dq$-operator , Acta Math., 113 (1965), 89–152.
• M. Klimek, Extremal plurisubharmonic functions and invariant pseudodistances , Bull. Soc. Math. France, 113 (1985), 231–240.
• S. Krantz and J. Yu, On the Bergman invariant and curvatures of the Bergman metric , Illinois J. Math., 40 (1996), 226–244.
• J. McNeal, Lower bounds on the Bergman metric near a point of finite type , Ann. Math., 136 (1992), 339–360.
• N. Nikolov, Localization of invariant metrics , Arch. Math., 79 (2002), 67–73.
• T. Ohsawa, Boundary behavior of the Bergman kernel function on pseudoconvex domains , Publ. R.I.M.S. Kyoto University, 20 (1984), 897–902.
• T. Ohsawa and K. Takegoshi, On the extension of $L^2$-holomorphic functions , Math. Z., 195 (1987), 197–204.
• N. Sibony, Une classe de domaines pseudoconvexes , Duke Math. J., 55 (1987), 299–319.
• J. Yu, Peak functions on weakly pseudoconvex domains , Indiana Univ. Math. J., 43 (1994), 1271–1295.