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Chen S. Classifying convex compact ancient solutions to the affine curve shortening flow[J]. Journal of Geometric Analysis, 2013, 25(2): 1075-1079.
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Angenent S. Formal asymptotic expansions for symmetric ancient ovals in mean curvature flow[J]. Networks and Heterogeneous Media, 2013, 8(1): 1-8.
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Philip Broadbridge Yand Peter Vassiliou Z. Symmetry, Integrability and Geometry: Methods and Applications SIGMA 7 (2011), 052, 19 pages The Role of Symmetry and Separation in Surface Evolution and Cu…. 2011.
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Robert Haslhofer · Or Hershkovits. Ancient solutions of the mean curvature flow. 2013.
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Paul Bryan · Janelle Louie. Classification of Convex Ancient Solutions to Curve Shortening Flow on the Sphere. 2014.
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Shibing Chen. Convex solutions to the power-of-mean curvature flow. 2012.
We consider an embedded convex compact ancient solution t to the curve shortening flow in R2. We prove that t is either a
family of contracting circles, which is a type I ancient solution, or a family of evolving Angenent ovals, which is of type II.
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Erbar M, Kuwada K, Sturm K, et al. On the Equivalence of the Entropic Curvature-Dimension Condition and Bochner\u0027s Inequality on Metric Measure Spaces[J]. Inventiones Mathematicae, 2013, 201(3): 993-1071.
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Z Qian · Huichun Zhang · Xiping Zhu. Sharp Spectral Gap and Li-Yau's Estimate on Alexandrov Spaces. 2011.
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Ketterer C. Cones over metric measure spaces and the maximal diameter theorem[J]. Journal de Mathématiques Pures et Appliquées, 2013, 103(5): 1228-1275.
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Jiang R. Cheeger-harmonic functions in metric measure spaces revisited[J]. Journal of Functional Analysis, 2013, 266(3): 1373-1394.
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Xia C. Local gradient estimate for harmonic functions on Finsler manifolds[J]. Calculus of Variations and Partial Differential Equations, 2013: 849-865.
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Jiang R, Koskela P. Isoperimetric inequality from the poisson equation via curvature[J]. Communications on Pure and Applied Mathematics, 2012, 65(8): 1145-1168.
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Jiang R, Zhang H. Hamilton\u0027s Gradient Estimates and A Monotonicity Formula for Heat Flows on Metric Measure Spaces[J]. Nonlinear Analysis-theory Methods \u0026 Applications, 2015: 32-47.
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Hua B, Xia C. A note on local gradient estimate on Alexandrov spaces[J]. Tohoku Mathematical Journal, 2013, 66(2): 259-267.
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J X Huang · Huichun Zhang. Harmonic Maps Between Alexandrov Spaces. 2016.
In this paper, we establish a Bochner-type formula on Alexandrov spaces with Ricci curvature bounded below. Yau’s gradient estimate for harmonic functions is also obtained on Alexandrov spaces.