We introduce the character of Thurston's circle packings in the hyperbolic background geometry.Consequently, some quite simple criteria are obtained for the existence of hyperbolic circle packings. For example,if a closed surface X admits a circle packing with all the vertex degrees d_(i)≥7, then it admits a unique complete hyperbolic metric so that the triangulation graph of the circle packing is isotopic to a geometric decomposition of X. This criterion is sharp due to the fact that any closed hyperbolic surface admits no triangulations with all d_(i)≤6. As a corollary, we obtain a new proof of the uniformization theorem for closed surfaces with genus g≥2;moreover, any hyperbolic closed surface has a geometric decomposition. To obtain our results, we use Chow-Luo's combinatorial Ricci flow as a fundamental tool.
This series is an engaging read about Chinese characters,explaining their origins and the logic behind their creation.It allows readers to grasp the essence of Chinese character culture and appreciate the ancient art of character creation.The series is divided into eight mini-volumes,each focusing on eight major categories of common everyday objects.It introduces over 260 key characters studied in elementary school and more than 200 practical idioms for elementary students.
