An Essay on the Bergman Metric and Balanced Domains

Abstract

Let Ω be a bounded pseudoconvex domain in ℂⁿ and let EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4samaaBa % aaleaacqqHPoWvaeqaaOGaaiikaiaadQhacaGGSaGabm4DayaaraGa % aiykaaaa!3CA4!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$${K_\Omega }(z,\bar w)$$ be the Bergman kernel function of Ω. The boundary behavior of K _Ω reflects the mass distribution of L ² holomorphic functions on Ω through the geometry of the boundary ∂Ω in a very natural way, as one can see it from [H1] and various subsequent works (cf. [D], [P], [D’A], [O-1,2], [D-H-O], [C], [D-H], [B-S-Y], etc.). From the viewpoint of biholomorphic geometry, the Bergman metric

EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizaiaado % hadaqhaaWcbaGaeuyQdCfabaGaaGOmaaaakiabg2da9maaqahabaWa % aSaaaeaacqGHciITdaahaaWcbeqaaiaaikdaaaGcciGGSbGaai4Bai % aacEgacaWGlbWaaSbaaSqaaiabfM6axbqabaGccaGGOaGaamOEaiaa % cYcaceWG6bGbaebacaGGPaaabaGaeyOaIyRaamOEamaaCaaaleqaba % GaeqySdegaaOGaeyOaIyRabmOEayaaraWaaWbaaSqabeaacqaHYoGy % aaaaaaqaaiabeg7aHjaacYcacqaHYoGycqGH9aqpcaaIXaaabaGaam % OBaaqdcqGHris5aOGaamizaiaadQhadaahaaWcbeqaaiabeg7aHbaa % kiabgEPielaadsgaceWG6bGbaebadaahaaWcbeqaaiabek7aIbaaaa % a!6248!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$ds_\Omega ^2 = \sum\limits_{\alpha ,\beta = 1}^n {\frac{{{\partial ^2}\log {K_\Omega }(z,\bar z)}}{{\partial {z^\alpha }\partial {{\bar z}^\beta }}}} d{z^\alpha } \otimes d{\bar z^\beta }$$

is also a natural quantity attached to Ω. Being a Hermitian metric invariant under the biholomorphic transformations, the Bergman metric is of intrinsic nature, while the values of K _Ω are not. It is well known that one can draw important information on proper holomorphic mappings from the asymptotics of K _Ω and ds ²_Ω (see [F] and [B-N] for more precise statements). Besides such an application, the boundary behavior of the Bergman kernel and the metric is of considerable significance in the current complex analysis, because they supply questions that urge further developments of the so called L ² method for the ∂-operator. For instance, it was asked in [O-1] whether or not EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4samaaBa % aaleaacqqHPoWvaeqaaOGaaiikaiaadQhacaGGSaGabmOEayaaraGa % aiykamrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbacfaGae8 % 3CIKOaeqiTdq2aaSbaaSqaaiabfM6axbqabaGccaGGOaGaamOEaiaa % cMcadaahaaWcbeqaaiabgkHiTiaaikdaaaaaaa!4F84!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$${K_\Omega }(z,\bar z) \gtrsim {\delta _\Omega }{(z)^{ - 2}}$$ if ∂Ωis C ²-smooth, δ_Ω (z) being the distance from z to ∂Ω,and this was answered affirmatively in [O-T] as a corollary of a very general extension theorem for L ² holomorphic functions, which found even an application to algebraic geometry (cf. [A-S]).