Intuition for Reinhardt domains

Question

I'm taking a complex analysis course, and in a lecture on several complex variables my professor defined Reinhardt domains in $\mathbb{C}^n$ as domains invariant under the action of the n-dimensional torus. I'd like help to better understand this concept. Some questions:

1. The only examples I can think of are either n-dimensional balls or shells. Are there any other Reinhardt domains?
2. My professor drew a box with the top right corner cut out (in the space of absolute values) and claimed this was also a Reinhardt domain in $\mathbb{C}^2.$ I don't see why that would be: aren't there rotations that would land part of this domain into that corner?
3. I don't yet understand why this would be a useful definition to make. Why is this the right concept for domains for functions of several variables? Any help would be appreciated

Answer 1

The subclass of complete Reinhardt domains (domains closed under the map $(z_1,\ldots z_n) \mapsto (\lambda_1z_1,\ldots, \lambda_nz_n)$ for $\lambda_i \in \mathbb{C}$ and $|\lambda_i| \leq 1$) are useful for two main reasons, in my opinion.

1. Among the easiest domains to do analysis on are those whose defining functions depend only on the modulus of the variables. For example the ball given by the defining function $r_1(z) = \sum |z_k|^2 - 1$ or an egg domain like $r_2(z) = \sum_{k = 1}^{n} |z_k|^2 + |z_{n + 1}|^2 - 1$ are usually much easier to work with than domains whose defining functions depend on the real part of a variable for instance. Moreover in two-dimensions they can be somewhat visualized as your professor described. These domains are complete Reinhardt and thus also Reinhardt. It is easy to come up with examples of complete Reinhardt domains. Take a polynomial $p$ in the variables $|z_1|,\ldots, |z_n|$ such that $p(0) = 0$ and then let $r(z) = p(|z_1|,\ldots, |z_n|) - 1$. Then check to make sure the axioms of a defining function are satisfied.

2. The domain of convergence for power series are complete Reinhardt domains and the pseudoconvex complete Reinhardt domains are those that are logarithmically convex. You can see Krantz's book, chapter 2, for an in depth discussion on these topics.