Rigidity of proper holomorphic mappings between equidimensional bounded symmetric domains

Abstract

Abstract. We prove that any proper holomorphic mapping between two equidimensional irreducible bounded symmetric domains with rank ≥ 2 is a biholomorphism. The proof of the main result in this paper will be achieved by a differential-geometric study of a special class of complex geodesic curves on the bounded symmetric domains with respect to their Bergman metrics.

Proper holomorphic maps between bounded symmetric domains revisited

“…In such a result, the condition rank(D 1 ) ≥ rank(D 2 ) ≥ 2 is indispensible. Adapting Tsai's ideas to the equidimensional case, Tu [19] established Theorem 1.1 in the higher-rank case, assuming D 1 is irreducible. In using Tsai's ideas, the assumption that D 1 is irreducible is essential -see [19,Proposition 3.3] -and it is not clear that a small mutation of those ideas allows one to weaken this assumption.…”

“…Adapting Tsai's ideas to the equidimensional case, Tu [19] established Theorem 1.1 in the higher-rank case, assuming D 1 is irreducible. In using Tsai's ideas, the assumption that D 1 is irreducible is essential -see [19,Proposition 3.3] -and it is not clear that a small mutation of those ideas allows one to weaken this assumption. In our work, we are able to assume either D 1 or D 2 to be irreducible precisely by not relying too heavily on the fine structure of these domains.…”

“…Indeed, we wish to emphasize that the focus of this work is not on mopping up the residual cases in Theorem 1.1. The methods in [6] (and [20], on which [6] relies), [18] and [19] are tied, in a rather maximalistic way, to the fine structure of a bounded symmetric domain. In contrast, we present some ideas that make very mild use of the underlying orbit structure of the bounded symmetric domains.…”

“…In contrast, apart from, and owing to, a result of Bell [2] on boundary behaviour, our proof involves rather "soft" methods. b) The techniques underlying [18] and [19] rely almost entirely on the fine structure of a bounded symmetric domain. Specifically, they involve studying the effect of a proper holomorphic map on the characteristic symmetric subspaces of a bounded symmetric domain of rank ≥ 2.…”

Proper slice regular functions over quaternions

“…Alexander 定理得到了广泛关注, 被推广到各种区域上, 如强拟凸域 [10] 、具有实解析边界的拟凸 域 [11,12] 、推广的椭球体 [13] 、秩大于 2 的不可约有界对称域 [14][15][16] 、有界圆型域 [17] 、平衡域 [18,19] 和 华域 [20] . 关于全纯逆紧映射的更多研究成果可以参见早期的综述文献 [21,22].…”