2018/01/01 by Kefeng Liu, Liu, Kefeng, Yunhui Wu +1
Mathematics · #Analytic and geometric function theory #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #math.CV #math.DG
paper · pdf · doi:10.48550/arxiv.1801.00459
36 pages, comments are welcome
openalex publication_date 2018/01/01 · arxiv created 2018/09/06 · arxiv updated 2018/09/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We first show that for a bounded pseudoconvex domain with a manifold quotient of finite-volume in the sense of Kahler-Einstein measure, the identity component of the automorphism group of this domain is semi-simple without compact factors. This partially answers an open question in [Fra95]. Then we apply this result in different settings to solve several open problems, for examples, (1). We prove that the automorphism group of the Griffiths domain [Gri71] in ℂ2 is discrete. This gives a complete answer to an open question raised four decades ago. (2). We show that for a contractible HHR/USq complex manifold D with a finite-volume manifold quotient M, if D contains a one-parameter group of holomorphic automorphisms and the fundamental group of M is irreducible, then D is biholomorphic to a bounded symmetric domain. This theorem can be viewed as a finite-volume version of Nadel-Frankel's solution for the Kahzdan conjecture, which has been open for years. (3). We show that for a bounded convex domain D⊂ ℂn of C2-smooth boundary, if D has a finite-volume manifold quotient with an irreducible fundamental group, then D is biholomorphic to the unit ball in ℂn, which partially solves an old conjecture of Yau. For (2) and (3) above, if the complex dimension is equal to 2, more refined results will be provided.