Parameterized combinatorial curvatures and parameterized combinatorial curvature flows for discrete conformal structures on polyhedral surfaces

Discrete conformal structure on polyhedral surfaces is a discrete analogue of the smooth that assigns metrics by scalar functions defined vertices. It unifies and generalizes tangential circle packing, Thurston's inversive distance packing vertex scaling described Luo [29] others. In this paper, we introduce combinatorial α-curvature for structures surfaces, which parameterized generalization classical curvature. Then prove local global rigidity with respect to confirms Glickenstein conjecture in [22]. To study Yamabe problem α-curvature, α-Ricci flow Chow-Luo's Ricci packings [4] Luo's surfaces. handle potential singularities flow, extend through extending inner angles triangles constants. Under existence prescribed curvature, solution extended proved exist all time converge exponentially fast any initial value. This another [22] convergence gives an almost equivalent characterization solvability terms provides effective algorithm finding α-curvatures.