A Bernstein type theorem of ancient solutions to the mean curvature flow
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- by Keita Kunikawa PDF
- Proc. Amer. Math. Soc. 144 (2016), 1325-1333 Request permission
Abstract:
We derive a curvature estimate for the entire graphic solution to the mean curvature flow. As a consequence we show a Bernstein type theorem for ancient solutions (or eternal solutions) to the mean curvature flow.References
- Chao Bao and Yuguang Shi, Gauss maps of translating solitons of mean curvature flow, Proc. Amer. Math. Soc. 142 (2014), no. 12, 4333–4339. MR 3267001, DOI 10.1090/S0002-9939-2014-12209-X
- Klaus Ecker and Gerhard Huisken, Mean curvature evolution of entire graphs, Ann. of Math. (2) 130 (1989), no. 3, 453–471. MR 1025164, DOI 10.2307/1971452
- R. Haslhofer and O. Hershkovits, Ancient solutions of the mean curvature flow, http://arxiv.org/abs/1308.4095 (2013), preprint.
- Gerhard Huisken, Flow by mean curvature of convex surfaces into spheres, J. Differential Geom. 20 (1984), no. 1, 237–266. MR 772132
- Gerhard Huisken and Carlo Sinestrari, Mean curvature flow singularities for mean convex surfaces, Calc. Var. Partial Differential Equations 8 (1999), no. 1, 1–14. MR 1666878, DOI 10.1007/s005260050113
- G. Huisken and C. Sinestrari, Convex ancient solutions of the mean curvature flow, http://arxiv.org/abs/1405.7509 (2014), preprint.
- K. Kunikawa, Bernstein-type theorem of translating solitons in arbitrary codimension with flat normal bundle, to appear in Calc. Var. Partial Differential Equations.
- Peter Li and Shing-Tung Yau, On the parabolic kernel of the Schrödinger operator, Acta Math. 156 (1986), no. 3-4, 153–201. MR 834612, DOI 10.1007/BF02399203
- Philippe Souplet and Qi S. Zhang, Sharp gradient estimate and Yau’s Liouville theorem for the heat equation on noncompact manifolds, Bull. London Math. Soc. 38 (2006), no. 6, 1045–1053. MR 2285258, DOI 10.1112/S0024609306018947
- Meng Wang, Liouville theorems for the ancient solution of heat flows, Proc. Amer. Math. Soc. 139 (2011), no. 10, 3491–3496. MR 2813381, DOI 10.1090/S0002-9939-2011-11170-5
- Lu Wang, A Bernstein type theorem for self-similar shrinkers, Geom. Dedicata 151 (2011), 297–303. MR 2780753, DOI 10.1007/s10711-010-9535-2
- Xu-Jia Wang, Convex solutions to the mean curvature flow, Ann. of Math. (2) 173 (2011), no. 3, 1185–1239. MR 2800714, DOI 10.4007/annals.2011.173.3.1
Additional Information
- Keita Kunikawa
- Affiliation: Mathematical Institute, Tohoku University, Sendai 980-8578, Japan
- Email: sb1m16@math.tohoku.ac.jp
- Received by editor(s): January 15, 2015
- Received by editor(s) in revised form: March 24, 2015
- Published electronically: August 11, 2015
- Communicated by: Lei Ni
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 1325-1333
- MSC (2010): Primary 53C44; Secondary 35C06
- DOI: https://doi.org/10.1090/proc/12802
- MathSciNet review: 3447682