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Complete \(\lambda \)-hypersurfaces of weighted volume-preserving mean curvature flow

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Abstract

In this paper, we introduce a special class of hypersurfaces which are called \(\lambda \)-hypersurfaces related to a weighted volume preserving mean curvature flow in the Euclidean space. We prove that \(\lambda \)-hypersurfaces are critical points of the weighted area functional for the weighted volume-preserving variations. Furthermore, we classify complete \(\lambda \)-hypersurfaces with polynomial area growth and \(H-\lambda \ge 0\). The classification result generalizes the results of Huisken (J Differ Geom 31:285–299, 1990) and Colding and Minicozzi (Ann Math 175:755–833, 2012).

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Acknowledgements

A part of this work was finished when the first author visited Beijing Normal University. We would like to express our gratitude to Professor Zizhou Tang and Dr. Wenjiao Yan for warm hospitality. We would also like to thank the referee for invaluable comments and suggestions.

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Authors and Affiliations

  1. Department of Applied Mathematics, Faculty of Sciences, Fukuoka University, 814-0180, Fukuoka, Japan

    Qing-Ming Cheng

  2. School of Mathematical Sciences, South China Normal University, 510631, Guangzhou, China

    Guoxin Wei

Authors
  1. Qing-Ming Cheng
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  2. Guoxin Wei
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Corresponding author

Correspondence to Guoxin Wei.

Additional information

Communicated by J. Jost.

Qing-Ming Cheng was partially supported by JSPS Grant-in-Aid for Scientific Research (B): No. 16H03937. Guoxin Wei was partially supported by NSFC Grant Nos. 11771154, 11371150.

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Cheng, QM., Wei, G. Complete \(\lambda \)-hypersurfaces of weighted volume-preserving mean curvature flow. Calc. Var. 57, 32 (2018). https://doi.org/10.1007/s00526-018-1303-4

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