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

Gregory Drugan (Oregon Episcopal School, Portland, Or., U.S.A.)

Weiyong He (Department of Mathematics, University of Oregon, Eugene, Or., U.S.A.)

Micah W. Warren (Department of Mathematics, University of Oregon, Eugene, Or., U.S.A.)

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.

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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