The complex geomety of a domain related to μ\mu-synthesis

We describe the basic complex geometry and function theory of the {\em pentablock} $\mathcal{P}$, which is the bounded domain in $\mathbb{C}^3$ given by $$\mathcal{P}= \{(a_{21}, \mathrm{tr} A, \det A): A= \begin{bmatrix} a_{ij}\end{bmatrix}_{i,j=1}^2 \in \mathbb{B}\}$$ where $\mathbb{B}$ denotes the open unit ball in the space of $2\times 2$ complex matrices. We prove several characterizations of the domain. We describe its distinguished boundary and exhibit a $4$-parameter group of automorphisms of $\mathcal{P}$. We show that $\mathcal{P}$ is intimately connected with the problem of $\mu$-synthesis for a certain cost function $\mu$ on the space of $2\times 2$ matrices defined in connection with robust stabilization by control engineers. We demonstrate connections between the function theories of $\mathcal{P}$ and $\mathbb{B}$. We show that $\mathcal{P}$ is polynomially convex and starlike.Comment: 36 pages, 2 figures. This version contains corrections of some inaccuracies and an expanded argument for Proposition 12.

A geometric characterization of the symmetrized bidisc

The symmetrized bidisc G def = {(z + w, zw) : |z| < 1, |w| < 1} has interesting geometric properties. While it has a plentiful supply of complex geodesics and of automorphisms, there is nevertheless a unique complex geodesic in G that is invariant under all automorphisms of G. Moreover, G is foliated by those complex geodesics that meet in one point and have nontrivial stabilizer. We prove that these properties, together with two further geometric hypotheses on the action of the automorphism group of G, characterize the symmetrized bidisc in the class of complex manifolds