arXiv subjects
Publications and source records attributed to Ze Zhou.
Thurston's Circle Pattern Theorem studies existence and rigidity of circle patterns of a given combinatorial type and the given non-obtuse exterior intersection angles. Using topological degree theory, variational principle, Teichmuller theory, and Sard's Theorem, this paper generalizes Circle Pattern Theorem to the case of obtuse exterior intersection angles.
This paper investigates circle patterns with obtuse exterior intersection angles on surfaces of finite topological type. We characterise the images of the curvature maps and establish several equivalent conditions regarding long time behaviors of Chow-Luo's combinatorial Ricci flows for these patterns. As consequences, several generalizations of circle pattern theorem are obtained. Moreover, our approach suggests a computational method to find the desired circle patterns.
This paper proves a deformation circle pattern theorem, which gives a complete description of those circle patterns with interstices in terms of the combinatorial type, the exterior intersections angles and the conformal structures of interstices. As results, the surface version of Rivin's theorem and the approximation property of packable surfaces are obtained.
A planar graph is inscribable if it is combinatorial equivalent to the skeleton of a polyhedra which is inscribed in a sphere. For an inscribable graph, in its combinatorial equivalent class, if we could always find polyhedra inscribed in any given convex surface which is sufficiently close to the sphere, then we call such an inscribable graph a stable one. By combining the Teichmüller theory of packings with differential topology method, in this paper there is investigation on the stability of some inscribable graphs.
The Midscribability Theorem, which was first proved by O. Schramm, states that: given a strictly convex body $K\subset\mathbb{R}^{3}$ with smooth boundary and a convex polyhedron $P$, there exists a polyhedron $Q \subset \mathbb{RP}^3$ combinatorially equivalent to $P$ which midscribes $K$. Here the word "midscribe" means that all it's edges are tangent to the boundary surface of $K$. By using of the intersection number technique, together with the Teichmüller theory of packings, this paper provides an alternative approach to this theorem. Furthermore, combining Schramm's method with the above ones, the authors prove a rigidity result concerning this theorem as well. Namely, such a polyhedron is unique under certain normalization conditions.