To extend Koebe-Andreev-Thurston's Circle Pattern Theorem, Cooper-Rivin proposed to study the deformation of triangulated 3-manifolds with tangential ball packings.
In this talk, we will follow Cooper-Rivin’s approach but with perpendicular ball packings which seem better to handle with. Let M be a triangulated three-dimensional manifold, we will define a combinatorial analogue of scalar curvature for M by ball packings. We will raise a combinatorial Yamabe problem and solve it under the assumption that all edge degrees are no bigger than 4. This is joint works with Wenshuai Jiang.

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