Circle packings in surfaces of finite type: an in situ approach with applications to moduli

Abstract

ROBERT BRINKS [lo, 1 l] used circle packings to parametrize the deformation space of a Kleinian group and applied his results to prove that the “circle packing points” (closed Riemann surfaces that can be filled by circle packings) form a dense subset of moduli space. In [9], the authors combined techniques of Brooks, Thurston [28], and BeardonStephenson [4, 5) to extend Brooks’ result to surfaces of finite conformal type, closed surfaces with a finite number of punctures. The methods there involve the in situ manipulation of circle packings and rely heavily on certain canonical Brooks’ packings of quadrilaterals as developed in [lo], along with canonical infinite packings of cusp regions. The latter are rigid, but the Brooks’ packings act like shock absorbers, permitting the small adjustments to modulus that lead to circle packing points. Our purposes here are threefold: first, to develop more fully and systematically the in situ approach to circle packings in hyperbolic surfaces begun in [9]; second, to define canonical packings of annuli that have a flexibility reminiscent of Brooks’ packings of quadrilaterals; and thiid, to apply these in an extension of Brooks’ result to surfaces of finite topological type, closed surfaces having a finite number of punctures and a finite number of half-annular ends. Our main result is the following.