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Paul Allen : The Dirichlet problem for curve shortening flow.

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We consider the Dirichlet problem for curve shortening flow on surfaces of constant curvature and show long-time existence of the flow when the initial curve is embedded in a convex region. Furthermore, the limit curve of the flow is a geodesic. The proof relies on an adaptation of Huisken's distance comparison estimate for planar curves, a maximum principle of Angenent, and a blow-up analysis of singularities.

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