Communications in Analysis and Geometry

Volume 28 (2020)

Number 8

The Second of Two Special Issues in Honor of Karen Uhlenbeck’s 75th Birthday

Special-Issue Editors: Georgios Daskalopoulos (Brown University), Kefeng Liu, Chuu-Lian Terng (U. of Cal. Irvine), and Shing-Tung Yau

Isotropic curve flows

Pages: 1807 – 1846



Chuu-Lian Terng (Department of Mathematics, University of California, Irvine, Calif., U.S.A.)

Zhiwei Wu (School of Mathematics, Sun Yat-sen University, Zhuhai, Guangdong, China)


A smooth curve $\gamma$ in $\mathbb{R}^{n+1,n}$ is isotropic if $\gamma , \gamma_x, \dotsc , \gamma^{(2n)}_x$ are linearly independent and the span of $\gamma , \gamma_x, \dotsc , \gamma^{(n−1)}_x$ is isotropic. We construct two hierarchies of isotropic curve flows on $\mathbb{R}^{n+1,n}$, whose differential invariants are solutions of Drinfeld–Sokolov’s KdV type soliton hierarchies associated to the affine Kac–Moody algebra $\hat{B}^{(1)}_n$ and $\hat{A}^{(2)}_{2n}$. For example, the $\hat{B}^{(1)}_1$-KdV is the KdV hierarchy and the $\hat{A}^{(2)}_2$-KdV hierarchy is the Kupershmidt–Kaup (KK) hierarchy. Hence we our study gives geometric interpretations of the KdV and KK equations as the curvature flows of natural geometric curve flows on the light cone of $\mathbb{R}^{2,1}$. Bi-Hamiltonian structures and conservation laws for isotropic curve flows on $\mathbb{R}^{n+1,n}$ are also given.

The authors’ research was supported in part by NSF of China Grant no. 11401327 and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.

Received 14 June 2018

Accepted 14 October 2019

Published 8 January 2021