双曲空间中曲线的逆曲率流研究
Study on Inverse Curvature Flows of Curves in Hyperbolic Space
摘要: 基于Tsai和Wang的线性插值流方法，重点推导双曲空间中凸曲线的逆曲率流下曲线长度和面积的演化方程，引入等周亏格并证明其单调性以刻画流的演化规律；通过构造与内球中心相关的支撑函数，建立曲率的边界估计，最终证明逆曲率流的长时存在性与收敛性。
Abstract: Based on the research method of Tsai and Wang, a linear interpolation flow model is constructed. We focus on deriving the evolution equations of curve length and area under the inverse curvature flow of convex curves in hyperbolic space, introduce the isoperimetric deficit and prove its monotonicity to characterize the evolution law of the flow. By constructing a support function related to the center of the inner sphere, the boundary estimate of curvature is established, and finally the long-time existence and convergence of the inverse curvature flow is proved.
文章引用：方博文. 双曲空间中曲线的逆曲率流研究[J]. 理论数学, 2026, 16(1): 93-104. https://doi.org/10.12677/pm.2026.161012

参考文献

[1] 潘生亮. 关于凸曲线的一些注记及其对曲率流的应用 [J]. 数学年刊 A 辑 (中文版),2000,(01):53-56.
[2] 潘生亮. 一般平面曲线流的一点注记 (英文)[J]. 数学研究,2000,(01):17-26.
[3] 高来源, 郝瑞霞, 潘生亮. 非局部平面曲线流 [J]. 中国科学: 数学,2024,54(03):407-422.
[4] L. Gao, S. Pan, and D. H. Tsai. On an area-preserving inverse curvature flow of convex closed plane curves[J]. Journal of Functional Analysis, vol. 280, no. 8, p. 108931, 2021.
[5] L. Gao, S. Pan, and D. H. Tsai. On a length-preserving inverse curvature flow of convex closed plane curves[J]. Journal of Differential Equation, vol. 269, no. 7, p. 5802-5831, 2020.
[6] Shengliang Pan, Juanna Yang. On a Non-Local Perimeter-Preserving Curve Evolution Problem for Convex Plane Curves[J], Manuscripta Mathematica, 2008, 127(4): 469-484.
[7] H. Guo and Z. Sun. On a family of inverse curvature flows for closed convex plane curves[J]. Nonlinear Analysis Real World Applications, vol. 50, pp. 1-7, 2019.
[8] Ben Andrews, Yong Wei. Quermassintegral Preserving Curvature Flow in Hyperbolic Space[J], Geometric and Functional Analysis, 2018, 28(5): 1183-1208.
[9] K. K. Kwong, Y. Wei, G. Wheeler, and V. M. Wheeler. On an inverse curvature flow in two-dimensional space forms[J]. Mathematische Annalen, vol. 384, no. 1/2, 2022.
[10] D. H. Tsai and X. L. Wang. On area-preserving and length-preserving nonlocal flow of convex closed plane curves[J]. 2014.
[11] H. Guo and S. Zhu. Linear interpolation on k(a)-type area-preserving and length-preserving curve flows[J]. Monat-shefte fur Mathematik, 2023.
[12] Y. Wei and B. Yang. Volume preserving flows for convex curves and surfaces in the hyperbolic space[J]. Journal of Functional Analysis, 2022.

为你推荐