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Bernstein theorem for translating solitons of hypersurfaces

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Abstract

In this paper, we prove a monotonicity formula and some Bernstein type results for translating solitons of hypersurfaces in \(\mathbb {R}^{n+1}\), giving some conditions under which a translating soliton is a hyperplane. We also show a gap theorem for the translating soliton of hypersurfaces in \(R^{n+k}\), namely, if the \(L^n\) norm of the second fundamental form of the soliton is small enough, then it is a hyperplane.

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Acknowledgements

The authors are very grateful to the unknown referees for helpful suggestions.

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Authors and Affiliations

  1. School of Mathematics and Physics, University of Science and Technology Beijing, 30 Xueyuan Road, Haidian District, Beijing, 100083, People’s Republic of China

    Li Ma

  2. Department of Mathematics, University of Valencia, Calle Dr. Moliner, 50, 46100, Burjassot, Valencia, Spain

    Vicente Miquel

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  1. Li Ma
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  2. Vicente Miquel
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Correspondence to Li Ma.

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The research is partially supported by the National Natural Science Foundation of China No. 11771124 and by the Project DGI (Spain) and FEDER Project MTM2016-77093-P. and the G.V. Project PROMETEOII/2014/064. This work was done when the first named author was visiting Valencia University in April 2014 and he would like to thank the hospitality of the Department of geometry and Topology.

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Ma, L., Miquel, V. Bernstein theorem for translating solitons of hypersurfaces. manuscripta math. 162, 115–132 (2020). https://doi.org/10.1007/s00229-019-01112-1

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  • DOI: https://doi.org/10.1007/s00229-019-01112-1

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