Holomorphic isometries into homogeneous bounded domains

TL;DR

This paper establishes two rigidity theorems for holomorphic isometries into homogeneous bounded domains, showing that certain Kähler-Ricci solitons are trivial and that these domains are not relatives of complex Euclidean spaces.

Contribution

It extends previous results by proving new rigidity theorems for holomorphic isometries and the non-relativity of homogeneous bounded domains with Euclidean spaces.

Findings

Kähler-Ricci solitons on homogeneous bounded domains are trivial (Kähler-Einstein).

Homogeneous bounded domains and complex Euclidean spaces are not relatives.

The results extend prior work by Loi, Mossa, Cheng, and Hao.

Abstract

We prove two rigidity theorems on holomorphic isometries into homogeneous bounded domains. The first shows that a K\"ahler-Ricci soliton induced by the homogeneous metric of a homogeneous bounded domain is trivial, i.e. K\"ahler-Einstein. In the second one we prove that a homogeneous bounded domain and the flat (definite or indefinite) complex Euclidean space are not relatives, i.e. they do not share a common K\"ahler submanifold (of positive dimension). Our theorems extend the results proved in [A. Loi, R. Mossa, Proc. Amer. Math. Soc. 149 (2021), no. 11, 4931-4941] and [X. Cheng, Y. Hao, Ann. Global Anal. Geom. 60 (2021), no. 1, 167-180] respectively.