Hsueh-Han HuangNational Tsing Hua UniversityNgai Hang ChanThe Chinese University of Hong KongKun ChenSouthwestern University of Finance and EconomicsChing-Kang IngNational Tsing Hua University
arXiv subject: Statistics Theory (math.ST)mathscidoc:2206.72001
Haslhofer R, Kleiner B. Mean curvature flow of mean convex hypersurfaces[J]. Communications on Pure and Applied Mathematics, 2013.
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Daskalopoulos P, Hamilton R, Sesum N, et al. Classification of compact ancient solutions to the Ricci flow on surfaces[C]., 2009.
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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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Broadbridge P, Vassiliou P J. The Role of Symmetry and Separation in Surface Evolution and Curve Shortening[J]. Symmetry Integrability and Geometry-methods and Applications, 2011.
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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.
Gigli N, Kuwada K, Ohta S, et al. Heat flow on Alexandrov spaces[J]. Communications on Pure and Applied Mathematics, 2010, 66(3): 307-331.
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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.
This is a personal update on some recent advances on the geometry of moduli spaces of Calabi–Yau manifolds, especially along the finite distance boundary with respect to the Weil–Petersson metric. Two main themes are metric completion and extremal transitions. Besides reviewing the known results, I will also raise some related questions.