https://neckpinch.com/authors/joseph-lauer/New arXiv papers by Joseph Lauer, from Neckpinch Tue, 10 Nov 2026 04:10:28 GMT https://neckpinch.com/papers/1601.05704/arxiv:1601.05704Thu, 21 Jan 2016 00:00:00 GMTJoseph Lauer — In this paper we prove that if γ is a Jordan curve on S² 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 S² 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 S².Curve shorteningWeak & singular flowsStability & uniqueness