Mathematical Research Letters

Volume 27 (2020)

Number 5

On the four-vertex theorem for curves on locally convex surfacessurfaces

Pages: 1261 – 1279



Shibing Chen (School of Mathematical Sciences, University of Science and Technology of China, Hefei, China)

Xu-Jia Wang (Centre for Mathematics and Its Applications, The Australian National University, Canberra, ACT, Australia)

Bin Zhou (School of Mathematical Science, Peking University, Beijing, China)


The classical four-vertex theorem describes a fundamental property of simple closed planar curves. It has been extended to space curves, namely a smooth, simple closed curve in $\mathbb{R}^3$ has at least four points with vanishing torsion if it lies on a convex surface. More recently, Ghomi [6] extended this property to curves lying on locally convex surfaces. In this paper we provide an alternative approach to the result via the theory of Monge–Ampère equations.

The second-named author was supported by ARC FL130100118 and DP170100929. The third-named author was supported by NSFC 11571018 and 11822101.

Received 11 December 2019

Accepted 9 May 2020

Published 12 January 2021