Curves of Finite Total Curvature

On the topology of the spaces of curvature constrained plane curves

It is well known that plane curves with the same endpoints are homotopic. An analogous claim for plane curves with the same endpoints and bounded curvature still remains open. In this work we find necessary and sufficient conditions for two plane curves with bounded curvature to be deformed, one to another, by a continuous one-parameter family of curves also having bounded curvature. We conclude that the space of these curves has either one or two connected components, depending on the distance between the endpoints. The classification theorem here presented answers a question raised in 1961 b…

球面背景几何中全测地曲率的预定组合 Ricci 流的收敛性

The Convergence of Prescribed Combinatorial Ricci Flows for Total Geodesic Curvatures in Spherical Background Geometry

针对球面背景几何中预定全测地曲率的圆模式度量问题，本文构造并分析预定组合 Ricci 流，证明其收敛性，并给出求解退化情形的两种方法。

$h$-原理：具有指定曲率的光滑曲线与纽结

$h$-Principles for smooth curves and knots with prescribed curvature

本文证明了具有指定曲率的光滑曲线满足$C^1$-稠密$h$-原理，从而在$R^3$中每个同痕类里都存在指定曲率的$C^\infty$纽结。

常曲率曲线的形变

Deformations of curves with constant curvature

该研究证明Rⁿ（n≥3）中常曲率曲线满足参数化C¹-稠密相对h原理，据此得出R³中常曲率纽结同痕或同伦的充要条件，并借助凸积分技术完成证明。

Total curvature and isotopy of graphs in $R^3$

Knot theory is the study of isotopy classes of embeddings of the circle $S^1$ into a 3-manifold, specifically $R^3$. The Fáry-Milnor Theorem says that any curve in $R^3$ of total curvature less than $4π$ is unknotted. More generally, a (finite) graph consists of a finite number of edges and vertices. Given a topological type of graphs $Γ$, what limitations on the isotopy class of $Γ$ are implied by a bound on total curvature? What does ``total curvature" mean for a graph? We define a natural notion of net total curvature of a graph $Γ$ in $R^3$, and prove that if $Γ$ is homeomorphic to the $θ$…

Total curvatures of holonomic links

A differential geometric characterization of the braid-index of a link is found. After multiplication by 2pi, it equals the infimum of the sum of total curvature and total absolute torsion over holonomic representatives of the link. Upper and lower bounds for the infimum of total curvature over holonomic representatives of a link are given in terms of its braid- and bridge-index. Examples showing that these bounds are sharp are constructed.