On the invariance of holomorphic mappings of the Hartogs domain over the minimal ball

In this paper, we study a family of generalized Hartogs type domain over the minimal ball, which is defined by the inequality ${\Vert z\Vert }^2+|z\cdot z|<{(1-{\Vert w\Vert }^2)}^p$, where $(z,w)\in C^n\times C^m$. We will show that the collection of all the holomorphic self-mappings of the Hartogs type domain over a minimal ball keeping the slice function invariant form a subgroup of the automorphism group. As an application, we can build a rigidity result for the automorphism group of the generalized Hartogs type domain over the minimal ball with $p=1$.