Umehara algebra and complex submanifolds of indefinite complex space forms

Abstract

The Umehara algebra is studied with motivation on the problem of the non-existence of common complex submanifolds. In this paper, we prove some new results in Umehara algebra and obtain some applications. In particular, if a complex manifolds admits a holomorphic polynomial isometric immersion to one indefinite complex space form, then it cannot admits a holomorphic isometric immersion to another indefinite complex space form of different type. Other consequences include the non-existence of the common complex submanifolds for indefinite complex projective space or hyperbolic space and a complex manifold with a distinguished metric, such as homogeneous domains, the Hartogs triangle, the minimal ball, the symmetrized polydisc, etc, equipped with their intrinsic Bergman metrics, which generalizes more or less all existing results.

arXiv commentarXiv admin note: text overlap with arXiv:1601.05907 by other authors

References in corpus (5)

• Bergman theory of certain generalized Hartogs triangles
• A bounded homogeneous domain and a projective manifold are not relatives
• Kähler immersions of Kähler-Ricci solitons into definite or indefinite complex space forms
• Holomorphic isometries into homogeneous bounded domains
• Submanifolds of some Hartogs domain and the complex Euclidean space