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# Communications in Analysis and Geometry

## Volume 26 (2018)

### Number 4

### Legendrian curve shortening flow in $\mathbb{R}^3$

Pages: 759 – 785

DOI: http://dx.doi.org/10.4310/CAG.2018.v26.n4.a4

#### Authors

#### Abstract

Motivated by Legendrian curve shortening flows in $\mathbb{R}^3$, we study the curve shortening flow of figure-eight curves in the plane. We show that, under some symmetry and curvature conditions, a figure-eight curve will shrink to a point at the first singular time. We also give a proof of short-time existence of Legendrian mean curvature flow in Sasaki–Einstein manifolds.

The second author was partially supported by NSF Grants DMS-1005392 and DMS-1611797.

The third author was partially supported by NSF Grant DMS-1438359.

Received 6 January 2016