# 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).

## Mathematics Subject Classification

53C44 53C42## Notes

### 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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