The central theme of this paper is the holomorphic spectral theory of the canonical Laplace operator of the complement of the “complexified unit circle” $\{(z,w) \in \widehat {{\mathbb C}}^2 \colon z \cdot w = 1\}$. We start by singling out a distinguished set of holomorphic eigenfunctions on the bidisk in terms of hypergeometric ${}_2F_1$ functions and prove that they provide a spectral decomposition of every holomorphic eigenfunction on the bidisk. As a second step, we identify the maximal domains of definition of these eigenfunctions and show that these maximal domains naturally determine the fine structure of the eigenspaces. Our main result gives an intrinsic classification of all closed Möbius invariant subspaces of eigenspaces of the canonical Laplacian of $\Omega $. Generalizing foundational prior work of Helgason and Rudin, this provides a unifying complex analytic framework for the real-analytic eigenvalue theories of both the hyperbolic and spherical Laplace operators on the open unit disk resp. the Riemann sphere and, in particular, shows how they are interrelated with one another.

In this paper, we will show that the unitary equivalence of two multiplication operators on the Bergman spaces on polygons depends on the geometry of the polygon.

For an analytic curve $\gamma :\,\left( a,\,b \right)\,\to \,\mathbb{C}$, the set of values approached by $\gamma \left( t \right)$, as $t\,\searrow \,\,a$ and as $t\,\nearrow \,b$ can be any two continua of $\mathbb{C}\,\cup \,\left\{ \infty \right\}$.

We consider a multi-parameter family of canonical coordinates and mirror maps originally introduced by Zudilin. This family includes many of the known one-variable mirror maps as special cases, in particular many of modular origin and the celebrated ‘quintic’ example of Candelas, de la Ossa, Green and Parkes. In a previous paper, we proved that all coefficients in the Taylor expansions at 0 of these canonical coordinates (and, hence, of the corresponding mirror maps) are integers. Here we prove that all coefficients in their Taylor expansions at 0 are positive. Furthermore, we provide several results about the behaviour of the canonical coordinates and mirror maps as complex functions. In particular, we address their analytic continuation, points of singularity, and radius of covergence. We present several very precise conjectures on the radius of covergence of the mirror maps and the sign pattern of the coefficients in their Taylor expansions at 0.