Analyzing the Wu metric on a class of eggs in ℂⁿ -- I

2015/03/10 by G. P. Balakumar, Balakumar, G. P., Prachi Mahajan +1
Mathematics · #Complex Variables (math.CV) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Holomorphic and Operator Theory

paper · pdf · doi:10.48550/arxiv.1503.02787

openalex publication_date 2015/03/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the Wu metric on convex egg domains of the form E_2m = \ z ∈ ℂⁿ : \vert z₁ \vert^2m + \vert z₂ \vert² + … + \vert z_n-1 \vert² + \vert z_n \vert² lt;1 \ where m ≥ 1/2, m ≠ 1. The Wu metric is shown to be real analytic everywhere except on a lower dimensional subvariety where it fails to be C²-smooth. Overall however, the Wu metric is shown to be continuous when m=1/2 and even C¹-smooth for each m>1/2, and in all cases, a non-Kähler Hermitian metric with its holomorphic curvature strongly negative in the sense of currents. This gives a natural answer to a conjecture of S. Kobayashi and H. Wu for such E_2m.

Citations

• Bounds for Invariant Distances on Pseudoconvex Levi Corank One Domains and Applications

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