Hostname: page-component-7dd5485656-bt4hw Total loading time: 0 Render date: 2025-10-31T03:49:15.965Z Has data issue: false hasContentIssue false
  • English
  • Français

ANCIENT SOLUTIONS OF CODIMENSION TWO SURFACES WITH CURVATURE PINCHING - RETRACTED

Part of: Parabolic equations and systems Global differential geometry

Published online by Cambridge University Press:  06 March 2020

ZHENGCHAO JI Open the ORCID record for ZHENGCHAO JI [Opens in a new window]
ZHENGCHAO JI*
Affiliation:
Center of Mathematical Sciences, Zhejiang University, Hangzhou, 310027, PR China email jizhengchao@zju.edu.cn
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We prove rigidity theorems for ancient solutions of geometric flows of immersed submanifolds. Specifically, we find conditions on the second fundamental form that characterise the shrinking sphere among compact ancient solutions for the mean curvature flow in codimension two surfaces.

Keywords

ancient solutionsmean curvature flow

MSC classification

Primary: 53C44: Geometric evolution equations (mean curvature flow, Ricci flow, etc.)
Secondary: 35K55: Nonlinear parabolic equations

Information

Type
Retraction
Information
Bulletin of the Australian Mathematical Society , Volume 102 , Issue 1 , August 2020 , pp. 162 - 171
Check for updates
Copyright
© 2020 Australian Mathematical Publishing Association Inc.

References

Andrews, B., ‘Noncollapsing in mean-convex mean curvature flow’, Geom. Topol. 16 (2012), 1413–1418.CrossRefGoogle Scholar
Andrews, B. and Baker, C., ‘Mean curvature flow of pinched submanifolds to spheres’, J. Differential Geom. 85 (2010), 357–396.CrossRefGoogle Scholar
Angenent, S., ‘Shrinking doughnuts’, Progr. Nonlinear Differential Equations Appl. 7 (1992), 21–38.Google Scholar
Angenent, S., ‘Formal asymptotic expansions for symmetric ancient ovals in mean curvature flow’, Netw. Heterog. Media 8 (2013), 1–8.CrossRefGoogle Scholar
Angenent, S., Daskalopoulos, P. and Sesum, N., ‘Unique asymptotics of ancient convex mean curvature flow solutions’, J. Diff. Geom. 3 (2019), 381–455.Google Scholar
Bakas, I. and Sourdis, C., ‘Dirichlet sigma models and mean curvature flow’, J. High Energy Phys. 06 (2007), 437–437.Google Scholar
Baker, C. and Nguyen, H. T., ‘Codimension two surfaces pinched by normal curvature evolving by mean curvature flow’, Ann. Inst. H. Poincaré Anal. Non Linéaire 34 (2017), 1599–1610.CrossRefGoogle Scholar
Bourni, T., Langford, M. and Tinaglia, G., ‘Deforming hypersurfaces of the sphere by their mean curvature’, Math. Z. 195 (1987), 205–219.Google Scholar
Brendle, S., Huisken, G. and Sinestrari, C., ‘Ancient solutions to the Ricci flow with pinched curvature’, Duke Math. J. 158 (2011), 537–551.CrossRefGoogle Scholar
Bryan, P., Ivaki, M. and Scheuer, J., ‘On the classification of ancient solutions to curvature flows on the sphere’, Preprint, arXiv:1604.01694, 2016.Google Scholar
Bryan, P. and Louie, J., ‘Classification of convex ancient solutions to curve shortening flow on the sphere’, J. Geom. Anal. 26 (2016), 858–872.CrossRefGoogle Scholar
Daskalopoulos, P., Hamilton, R. and Sesum, N., ‘Classification of compact ancient solutions to the curve shortening flow’, J. Differential Geom. 84 (2010), 455–464.CrossRefGoogle Scholar
Hamilton, R., ‘Formation of singularities in the Ricci flow’, Surv. Differ. Geom. 2 (1996), 7–136.CrossRefGoogle Scholar
Hamilton, R. and Hershkovits, O., ‘Ancient solutions of the mean curvature flow’, Commun. Anal. Geom. 24 (2016), 593–604.Google Scholar
Haslhofer, R. and Kleiner, B., ‘Mean curvature flow of mean convex hypersurfaces’, Commun. Pure Appl. Math. 70 (2017), 511–546.CrossRefGoogle Scholar
Huisken, G. and Sinestrari, C., ‘Convexity estimates for mean curvature flow and singularities of mean convex surfaces’, Acta. Math. 183 (1999), 45–70.CrossRefGoogle Scholar
Huisken, G. and Sinestrari, C., ‘Convex ancient solutions of the mean curvature flow’, J. Differential Geom. 101 (2015), 267–287.CrossRefGoogle Scholar
Langford, M., ‘A general pinching principle for mean curvature flow and applications’, Calc. Var. Partial Differential Equations 56 (2017), article ID 107.CrossRefGoogle Scholar
Langford, M. and Lynch, S., ‘Sharp one-sided curvature estimates for fully nonlinear curvature flows and applications to ancient solutions’, Preprint, arXiv:1704.03802, 2017.Google Scholar
Liu, K. F., Xu, H. W., Ye, F. and Zhao, E. T., ‘The extension and convergence of mean curvature flow in higher codimension’, Trans. Amer. Math. Soc. 370 (2017), 223–2261.Google Scholar
Lukyanov, S., Vitchev, E. and Zamolodchikov, A., ‘Integrable model of boundary interaction: the paper clip’, J. Nucl. Phys. B 683 (2004), 423–454.CrossRefGoogle Scholar
Risa, S. and Sinestrari, C., ‘Ancient solutions of geometric flows with curvature pinching’, J. Geom. Anal. 29 (2017), 1206–1232.CrossRefGoogle Scholar
White, B., ‘The nature of singularities in mean curvature flow of mean-convex sets’, J. Amer. Math. Soc. 16 (2003), 123–138.CrossRefGoogle Scholar