Summary: COMBINATORIAL SCALAR CURVATURE 1
Combinatorial scalar curvature and
rigidity of ball packings
By Daryl Cooper and Igor Rivin*
Let M 3 be a triangulated threedimensional manifold. In this paper we
define a combinatorial analogue of scalar curvature for M 3 ; and also a combi
natorial analogue of conformal deformation of the metric. We further define a
functional S on the combinatorial conformal deformation space, show that S is
concave, and show that critical points of S correspond precisely to metrics of
constant combinatorial scalar curvature on M 3 : These results are then applied
to showing rigidity of ball packings with prescribed combinatorics (the con
cepts are quite similar to Colin de Verdi`ere's work on circle packing of surfaces
. See also  for a related variational argument).
The plan of the paper is as follows. In section 1 we define the class
of conformal simplices in E 3 , and prove the necessary local versions of our
results. In section 2 we extend these techniques to conformal simplices in H 3 .
In Section 3 we study the deformation space of conformal simplices. In section
4 we prove the results on scalar curvature alluded to above, and in section 5
we discuss the ballpacking results.