The evolution of Jordan curves on \mathbbS² by curve shortening flow

2016/01/21 by Lauer, Joseph
#Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #General Topology (math.GN)

paper · doi:10.48550/arxiv.1601.05704

Abstract

In this paper we prove that if γ is a Jordan curve on \mathbbS² then there is a smooth curve shortening flow defined on (0,T) which converges to γ in C⁰ as t→ 0⁺ . Another perspective is that the level-set flow of γ is smooth. This is a generalization of the author's previous work where the planar case was studied. If a Jordan curve on \mathbbS² has Lebesgue measure zero then we show that the level-set flow instantly becomes a smooth closed curve. If the Lebesgue measure is positive then for small time the level-set flow is an annulus with smooth boundary. This second case should be interpreted as a failure of uniqueness. As in the planar case a key step in the proof is establishing a length estimate for smooth curves that depends on a geometric quantity called the r-multiplicity. The majority of this paper concerns the extension of this length estimate to \mathbbS².

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