Incompressibility of domain-filling circle packings (Q2402913)

It is known that the theory of circle packings of finite domains can be considered as a discrete analogue of the theory of complex analytic functions. For example, discrete versions of the Riemann mapping theorem and the uniformization theorem are known. Much less is known about discrete versions of theorems on existence and uniqueness of conformal mappings. The conformal modulus of quadrilaterals is a fundamental concept in conformal geometry. So, it is interesting to study circle packings of quadrilaterals. More precisely, a Jordan quadrilateral \(G=G(\alpha,\beta,\gamma,\delta)\) is a Jordan domain whose boundary is split into four arcs \(\alpha,\beta,\gamma,\delta\). A circle packing \(\mathcal{P}\) filling the quadrilateral \(G\) is a finite collection of circles which are contained in the closure of \(G\), and touch each other as well as the four boundary arcs. The tangency structure of the circle packing \(\mathcal{P}\) is encoded in a simplicial 2-complex \(\mathcal{Q}\). If all faces of \(\mathcal{Q}\) are triangles, \(\mathcal{Q}\) is irreducible. Let \(\mathcal{P}\) and \(\mathcal{P}'\) be circle packings filling the quadrilaterals \(G=G(\alpha,\beta,\gamma,\delta)\) and \(G'=G(\alpha',\beta',\gamma',\delta')\) with \(G'\subset G\), \(\beta'\subset \beta\), and \(\delta'\subset \delta\). Then it is proved that if \(\mathcal{P}\) and \(\mathcal{P}'\) have the same 2-complex, then \(\mathcal{P}=\mathcal{P}'\). Informally, this means that the circle packing \(\mathcal{P}\) is incompressible. Using the concept of prime ends this result can be generalized to arbitrary simply connected domains and more general circle ensembles (so called circle agglomerations). As a corollary it is proved that the circle agglomeration filling given trilateral and having given 2-comlex is unique. Also some more general results about existence and uniqueness of circle packings are announced. These results can be considered as a discrete version of a well-known Carathéodory theorem.