Projective and Conformal Rigidity of Circle Packings on Surfaces

Abstract: A still stunning theorem of Koebe, rediscovered by Thurston (noting the relationship to work of Andreev) asserts that each topological circle packing on a planar domain may be realized geometrically by round circles.   Kojima-Mozushima-Tan noted that complex projective structures on Riemann surfaces conjecturally formed a natural setting for this realization problem.

 

We make progress towards this conjecture, showing that the space of circle packings consistent with a given triangulation on a surface of genus at least two is projectively rigid, i.e. a packing on a complex projective surface is not deformable within that complex projective structure. More broadly, we show that the space of circle packings is a submanifold within the space of complex projective structures on that surface.  We describe some work towards the expected 'conformal rigidity theorem'  Joint work with Francesco Bonsante.