Abstract
We construct the ancient solutions of the hypersurface flows in Euclidean spaces studied by B. Andrews in 1994.
As time
Funding source: Simons Foundation
Award Identifier / Grant number: 229727
Award Identifier / Grant number: 581101
Funding source: National Natural Science Foundation of China
Award Identifier / Grant number: 11426195
Funding statement: Peng Lu is partially supported by Simons Foundation through Collaboration Grant 229727 and 581101. Jiuru Zhou is partially supported by a PRC grant NSFC 11426195.
Acknowledgements
The authors would like to thank Mat Langford for bringing our attention to [9] and [10]. Jiuru Zhou would like to thank China Scholarship Council for providing a fellowship as a visiting scholar at University of Oregon.
References
[1] S. J. Altschuler and L. F. Wu, Translating surfaces of the non-parametric mean curvature flow with prescribed contact angle, Calc. Var. Partial Differential Equations 2 (1994), no. 1, 101–111. 10.1007/BF01234317Suche in Google Scholar
[2] B. Andrews, Contraction of convex hypersurfaces in Euclidean space, Calc. Var. Partial Differential Equations 2 (1994), no. 2, 151–171. 10.1007/BF01191340Suche in Google Scholar
[3] B. Andrews, Fully nonlinear parabolic equations in two space variables, preprint (2004), https://arxiv.org/abs/math/0402235v1. Suche in Google Scholar
[4] B. Andrews, Pinching estimates and motion of hypersurfaces by curvature functions, J. reine angew. Math. 608 (2007), 17–33. 10.1515/CRELLE.2007.051Suche in Google Scholar
[5] B. Andrews, M. Langford and J. McCoy, Non-collapsing in fully non-linear curvature flows, Ann. Inst. H. Poincaré Anal. Non Linéaire 30 (2013), no. 1, 23–32. 10.1016/j.anihpc.2012.05.003Suche in Google Scholar
[6] B. Andrews, J. McCoy and Y. Zheng, Contracting convex hypersurfaces by curvature, Calc. Var. Partial Differential Equations 47 (2013), no. 3–4, 611–665. 10.1007/s00526-012-0530-3Suche in Google Scholar
[7] S. Angenent, Formal asymptotic expansions for symmetric ancient ovals in mean curvature flow, Netw. Heterog. Media 8 (2013), no. 1, 1–8. 10.3934/nhm.2013.8.1Suche in Google Scholar
[8] S. Angenent, P. Daskalopoulos and N. Sesum, Uniqueness of two-convex closed ancient solutions to the mean curvature flow, preprint (2018), https://arxiv.org/abs/1804.07230. 10.4007/annals.2020.192.2.2Suche in Google Scholar
[9] T. Bourni and M. Langford, Type-II singularities of two-convex immersed mean curvature flow, Geom. Flows 2 (2017), no. 1, 1–17. 10.1515/geofl-2016-0001Suche in Google Scholar
[10] T. Bourni, M. Langford and G. Tinglia, Collapsing ancient solutions of mean curvature flow, preprint (2017), https://arxiv.org/abs/1705.06981. 10.4310/jdg/1632506300Suche in Google Scholar
[11] S. Brendle and G. Huisken, A fully nonlinear flow for two-convex hypersurfaces in Riemannian manifolds, Invent. Math. 210 (2017), no. 2, 559–613. 10.1007/s00222-017-0736-2Suche in Google Scholar
[12] P. Daskalopoulos, R. Hamilton and N. Sesum, Classification of compact ancient solutions to the curve shortening flow, J. Differential Geom. 84 (2010), no. 3, 455–464. 10.4310/jdg/1279114297Suche in Google Scholar
[13] P. Daskalopoulos, R. Hamilton and N. Sesum, Classification of ancient compact solutions to the Ricci flow on surfaces, J. Differential Geom. 91 (2012), no. 2, 171–214. 10.4310/jdg/1344430821Suche in Google Scholar
[14] K. Ecker, Regularity theory for mean curvature flow, Progr. Nonlinear Differential Equations Appl. 57, Birkhäuser, Boston 2004. 10.1007/978-0-8176-8210-1Suche in Google Scholar
[15] R. Haslhofer and O. Hershkovits, Ancient solutions of the mean curvature flow, Comm. Anal. Geom. 24 (2016), no. 3, 593–604. 10.4310/CAG.2016.v24.n3.a6Suche in Google Scholar
[16] D. Hoffman, T. Ilmanen, F. Martín and B. White, Graphical translators for mean curvature flow, Calc. Var. Partial Differential Equations 58 (2019), no. 4, Paper No. 117. 10.1007/s00526-019-1560-xSuche in Google Scholar
[17] N. V. Krylov, Nonlinear elliptic and parabolic equations of the second order, Math. Appl. (Sov. Ser.) 7, D. Reidel Publishing, Dordrecht 1987. 10.1007/978-94-010-9557-0Suche in Google Scholar
[18] G. Perelman, The entropy formula for the Ricci ow and its geometric applications, preprint (2002), https://arxiv.org/abs/math/0211159. Suche in Google Scholar
[19] X.-J. Wang, Convex solutions to the mean curvature flow, Ann. of Math. (2) 173 (2011), no. 3, 1185–1239. 10.4007/annals.2011.173.3.1Suche in Google Scholar
[20] B. White, The nature of singularities in mean curvature flow of mean-convex sets, J. Amer. Math. Soc. 16 (2003), no. 1, 123–138. 10.1090/S0894-0347-02-00406-XSuche in Google Scholar
© 2020 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- Virtual cycles of gauged Witten equation
- Arakelov geometry on degenerating curves
- Ancient solutions for Andrews’ hypersurface flow
- On the regular-convexity of Ricci shrinker limit spaces
- The metric geometry of singularity types
- Profinite rigidity for twisted Alexander polynomials
- On the set of divisors with zero geometric defect
- Smooth rational affine varieties with infinitely many real forms
Artikel in diesem Heft
- Frontmatter
- Virtual cycles of gauged Witten equation
- Arakelov geometry on degenerating curves
- Ancient solutions for Andrews’ hypersurface flow
- On the regular-convexity of Ricci shrinker limit spaces
- The metric geometry of singularity types
- Profinite rigidity for twisted Alexander polynomials
- On the set of divisors with zero geometric defect
- Smooth rational affine varieties with infinitely many real forms