Some remarks on the Kobayashi--Fuks metric on strongly pseudoconvex domains

2021/10/31 by Diganta Borah, Debaprasanna Kar
Mathematics · #Bergman kernel #Bounded function #Curvature #Diagonal #Domain (mathematical analysis) #Geometry #Geometry and complex manifolds #Holomorphic and Operator Theory #Holomorphic function #Mathematical analysis #Mathematics #Meromorphic and Entire Functions #Metric (unit) #Pure mathematics #Ricci curvature #math.CV #msc:32A25 #msc:32A36 #msc:32F45

paper · pdf · doi:10.1016/j.jmaa.2022.126162

published as Journal of Mathematical Analysis and Applications 2022 · 23 pages

openalex created_date 2021/10/25 · openalex publication_date 2022/03/10 · arxiv created 2022/03/14 · arxiv updated 2022/03/15 · openalex updated_date 2026/08/05

Abstract

The Ricci curvature of the Bergman metric on a bounded domain D⊂ ℂⁿ is strictly bounded above by n+1 and consequently log (K_Dⁿ⁺¹g_B,D), where K_D is the Bergman kernel for D on the diagonal and g_{B, D} is the Riemannian volume element of the Bergman metric on D, is the potential for a Kähler metric on D known as the Kobayashi--Fuks metric. In this note we study the localization of this metric near holomorphic peak points and also show that this metric shares several properties with the Bergman metric on strongly pseudoconvex domains.

Citations

• L 2 -Cohomology and Index Theorem for the Bergman Metric
• Narasimhan--Simha type metrics on strongly pseudoconvex domains in ℂⁿ