Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter May 21, 2016

A characterization of the grim reaper cylinder

  • Francisco Martín EMAIL logo , Jesús Pérez-García , Andreas Savas-Halilaj and Knut Smoczyk

Abstract

In this article we prove that a connected and properly embedded translating soliton in 3 with uniformly bounded genus on compact sets which is C1-asymptotic to two planes outside a cylinder, either is flat or coincide with the grim reaper cylinder.

Funding source: MINECO-FEDER

Award Identifier / Grant number: MTM2014-52368-P

Award Identifier / Grant number: BES-2012-055302

Award Identifier / Grant number: SM 78/6-1

Funding statement: F. Martín and J. Pérez-García are partially supported by MINECO-FEDER grant no. MTM2014-52368-P. J. Pérez-García is also supported by MINECO (FPI grant, BES-2012-055302) and A. Savas-Halilaj and K. Smoczyk by DFG SM 78/6-1.

Acknowledgements

The authors would like to thank Brian White and Antonio Ros for many stimulating and helpful conversations. Moreover, the authors would also like to thank Oliver Schnürer and Leonor Ferrer for plenty of useful discussions. Finally, we would like to thank the referee for the valuable comments on the content of the manuscript and the suggestions for improving the paper.

References

[1] A. D. Alexandrov, Uniqueness theorems for surfaces in the large, Vestnik Leningrad Univ. Math. 11 (1956), 5–17. Search in Google Scholar

[2] H. I. Choi and R. Schoen, The space of minimal embeddings of a surface into a three-dimensional manifold of positive Ricci curvature, Invent. Math. 81 (1985), 387–394. 10.1007/BF01388577Search in Google Scholar

[3] J. Dávila, M. del Pino and X.-H. Nguyen, Finite topology self-translating surfaces for the mean curvature flow in 3, preprint (2015), http://arxiv.org/abs/1501.03867. Search in Google Scholar

[4] T. Ilmanen, Elliptic regularization and partial regularity for motion by mean curvature, Mem. Amer. Math. Soc. 108 (1994), no. 520. 10.1090/memo/0520Search in Google Scholar

[5] F. Martín, A. Savas-Halilaj and K. Smoczyk, On the topology of translating solitons of the mean curvature flow, Calc. Var. Partial Differential Equations 54 (2015), 2853–2882. 10.1007/s00526-015-0886-2Search in Google Scholar

[6] X.-H. Nguyen, Translating tridents, Comm. Partial Differential Equations 34 (2009), 257–280. 10.1080/03605300902768685Search in Google Scholar

[7] X.-H. Nguyen, Complete embedded self-translating surfaces under mean curvature flow, J. Geom. Anal. 23 (2013), 1379–1426. 10.1007/s12220-011-9292-ySearch in Google Scholar

[8] X.-H. Nguyen, Doubly periodic self-translating surfaces for the mean curvature flow, Geom. Dedicata 174 (2015), 177–185. 10.1007/s10711-014-0011-2Search in Google Scholar

[9] J. Pérez and A. Ros, Properly embedded minimal surfaces with finite total curvature, The global theory of minimal surfaces in flat spaces (Martina Franca 1999), Lecture Notes in Math. 1775, Springer, Berlin (2002), 15–66. 10.1007/978-3-540-45609-4_2Search in Google Scholar

[10] R. M. Schoen, Uniqueness, symmetry, and embeddedness of minimal surfaces, J. Differential Geom. 18 (1983), 791–809. 10.4310/jdg/1214438183Search in Google Scholar

[11] G. Smith, On complete embedded translating solitons of the mean curvature flow that are of finite genus, preprint (2015), http://arxiv.org/abs/1501.04149. Search in Google Scholar

[12] B. White, Evolution of curves and surfaces by mean curvature, Proceedings of the international congress of mathematicians – ICM 2002. Vol. I: Plenary lectures and ceremonies (Beijing 2002), Higher Education Press, Beijing (2002), 525–538. Search in Google Scholar

[13] B. White, On the compactness theorem for embedded minimal surfaces in 3-manifolds with locally bounded area and genus, preprint (2015), http://arxiv.org/abs/1503.02190v1. 10.4310/CAG.2018.v26.n3.a7Search in Google Scholar

[14] B. White, Controlling area blow-up in minimal or bounded mean curvature varieties, J. Differential Geom. 102 (2016), no. 3, 501–535. 10.4310/jdg/1456754017Search in Google Scholar

Received: 2015-08-24
Revised: 2015-12-21
Published Online: 2016-05-21
Published in Print: 2019-01-01

© 2018 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 7.2.2023 from https://www.degruyter.com/document/doi/10.1515/crelle-2016-0011/html
Scroll Up Arrow