Any smooth closed immersed curve in R^n with a one-to-one convex projection onto a 2-plane develops a Type I singularity and becomes asymptotically circular under curve shortening flow, enabling a perturbation result that is an analog of Huisken's conjecture.
Lectures on mean curvature flows in higher codimensions
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abstract
Mean curvature flows of hypersurfaces have been extensively studied and there are various different approaches and many beautiful results. However, relatively little is known about mean curvature flows of submanifolds of higher codimensions. This notes starts with some basic materials on submanifold geometry, and then introduces mean curvature flows in general dimensions and co-dimensions. The related techniques in the so called "blow-up" analysis are also discussed. At the end, we present some global existence and convergence results for mean curvature flows of two-dimensional surfaces in four-dimensional ambient spaces.
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math.DG 1years
2025 1verdicts
UNVERDICTED 1representative citing papers
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Singularities of Curve Shortening Flow with Convex Projections
Any smooth closed immersed curve in R^n with a one-to-one convex projection onto a 2-plane develops a Type I singularity and becomes asymptotically circular under curve shortening flow, enabling a perturbation result that is an analog of Huisken's conjecture.