反逆球堆度量的形变 III

Given a triangulated surface M, we use Ge–Xu's α-flow [10] to deform any initial inversive distance circle packing metric to a metric with constant α-curvature. More precisely, we prove that the inversive distance circle packing with constant α -curvature is unique if αχ(M)≤0, which generalizes Andreev–Thurston's rigidity results for circle packing with constant cone angles. We further prove that the solution to Ge–Xu's α-flow can always be extended to a solution that exists for all time and converges exponentially fast to constant α-curvature. Finally, we give some combinatorial and topological obstacles for the existence of constant α-curvature metrics.

给定一个三角剖分曲面\(M\)，我们利用葛 - 徐的\(\alpha\)-流\([10]\)将任何初始的反演距离圆填充度量变形为具有常\(\alpha\)-曲率的度量。更确切地说，我们证明了如果\(\alpha\chi(M)\leq0\)，则具有常\(\alpha\)-曲率的反演距离圆填充是唯一的，这推广了安德烈耶夫 - 瑟斯顿关于具有常锥角的圆填充的刚性结果。我们进一步证明，葛 - 徐的\(\alpha\)-流的解总是可以延拓为一个对所有时间都存在的解，并且以指数级快速收敛到常\(\alpha\)-曲率。最后，我们给出了一些关于常\(\alpha\)-曲率度量存在性的组合和拓扑障碍。