Extremal disc packings in compact hyperbolic surfaces
The author introduces extremal \(k\)-packings of compact Riemann surfaces as a natural generalization of extremal discs. A compact Riemann surface of genus \(g\geq 2\) is called a \(k\)-extremal surface if it contains an extremal \(k\)-packing, namely, a collection of \(k\) metric discs of the largest radius \(R=R(g,k)\) such that the discs are pairwise disjoint and each of them is homeomorphic to a topological disc. The radius \(R\) is described by \(g\) and \(k\) explicitly. For \(k\)-extremal surfaces of genus \(g\), the author obtains some interesting properties: the relation between \(k\) and \(g\) which \(k\)-extremal surfaces of genus \(g\) fulfill, a characterization of a surface group uniformizing a \(k\)-extremal surface as a subgroup of Fuchsian triangle groups of a certain type, determination of pairs \(\{k_1,k_2\}\) such that a Riemann surface of genus \(g\) is \(k_1\)-extremal and \(k_2\)-extremal simultaneously, and the relation between \(k\) and \(g\) such that extremal \(k\)-packings in \(k\)-extremal surfaces are unique or not. The author also shows that quasiplatonic surfaces do not contain extremal discs, that is, they are not (1-)extremal surfaces while there are infinitely many \(k\)'s for a surface to be quasiplatonic and \(k\)-extremal. Moreover, the actions of the automorphism groups on the extremal \(k\)-packings are explained by giving examples of compact Riemann surfaces.
scientific article; zbMATH DE number 6879859
scientific article