Classification of convex ancient solutions to curve shortening flow on the sphere

DOI10.1007/s12220-015-9574-xzbMath1338.53093arXiv1408.5523OpenAlexW2068072598MaRDI QIDQ282789

Paul Bryan, Janelle Louie

Publication date: 12 May 2016

Published in: The Journal of Geometric Analysis (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1408.5523

zbMATH Keywords

Harnack inequalityancient solutionsAleksandrov reflectioncurve shortening flow

Mathematics Subject Classification ID

Nonlinear parabolic equations (35K55) Heat and other parabolic equation methods for PDEs on manifolds (58J35)

Related Items (18)

Ancient solutions of codimension two surfaces with curvature pinching in $mathbb{R}^4$ ⋮ Ancient gradient flows of elliptic functionals and Morse index ⋮ Ancient solutions of geometric flows with curvature pinching ⋮ A Schur's theorem via a monotonicity and the expansion module ⋮ Convex ancient solutions to curve shortening flow ⋮ Collapsing and noncollapsing in convex ancient mean curvature flow ⋮ Sharp one-sided curvature estimates for fully nonlinear curvature flows and applications to ancient solutions ⋮ Aleksandrov reflection for extrinsic geometric flows of Euclidean hypersurfaces ⋮ Strong spherical rigidity of ancient solutions of expansive curvature flows ⋮ Classification of convex ancient free-boundary curve-shortening flows in the disc ⋮ On the Discrete Spectrum of Robin Laplacians in Conical Domains ⋮ ANCIENT SOLUTIONS OF CODIMENSION TWO SURFACES WITH CURVATURE PINCHING - RETRACTED ⋮ Eigenvalue counting function for Robin Laplacians on conical domains ⋮ Soliton solutions to the curve shortening flow on the sphere ⋮ Collapsing ancient solutions of mean curvature flow ⋮ Ancient solution of mean curvature flow in space forms ⋮ Ancient mean curvature flows out of polytopes ⋮ Ancient solutions for flow by powers of the curvature in \(\mathbb{R}^2\)

Cites Work

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• The heat equation shrinks embedded plane curves to round points
• Shortening embedded curves
• Asymptotic behavior of anisotropic curve flows
• A distance comparison principle for evolving curves
• Geometric aspects of Aleksandrov reflection and gradient estimates for parabolic equations
• Aleksandrov reflection and nonlinear evolution equations. I: The \(n\)-sphere and \(n\)-ball
• Aleksandrov reflection and geometric evolution of hypersurfaces

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