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

- 15 Downloads

## 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.

### References

- 1.Abresch, U., Langer, J.: The normalized curve shortening flow and homothetic solutions. J. Differ. Geom.
**23**, 175–196 (1986)MathSciNetCrossRefMATHGoogle Scholar - 2.Angenent, S.: Shrinking doughnuts. In: Lloyd, N.G., Ni, W.M., Peletier, L.A., Serrin, J. (eds.) Nonlinear Diffusion Equations and Their Equilibrium States, vol. 7, pp. 21–38. Birkhaüser, Boston, Basel, Berlin (1992)CrossRefGoogle Scholar
- 3.Barbosa, J.L., do Carmo, M.: Stability of hypersurfaces with constant mean curvature. Math. Z.
**185**, 339–353 (1984)MathSciNetCrossRefMATHGoogle Scholar - 4.Cao, H.-D., Li, H.: A gap theorem for self-shrinkers of the mean curvature flow in arbitrary codimension. Calc. Var. Partial Differ. Equ.
**46**, 879–889 (2013)MathSciNetCrossRefMATHGoogle Scholar - 5.Chang, J.-E.: One dimensional solutions of the \(\lambda \)-self shrinkers. arXiv:1410.1782
- 6.Cheng, Q.-M., Peng, Y.: Complete self-shrinkers of the mean curvature flow. Calc. Var. Partial Differ. Equ.
**52**, 497–506 (2015)MathSciNetCrossRefMATHGoogle Scholar - 7.
- 8.Colding, T.H., Minicozzi II, W.P.: Generic mean curvature flow I: generic singularities. Ann. Math.
**175**, 755–833 (2012)MathSciNetCrossRefMATHGoogle Scholar - 9.Ding, Q., Xin, Y.L.: The rigidity theorems of self shrinkers. Trans. Am. Math. Soc.
**366**, 5067–5085 (2014)MathSciNetCrossRefMATHGoogle Scholar - 10.Drugan, G.: An immersed \(S^2\) self-shrinker. Trans. Am. Math. Soc.
**367**, 3139–3159 (2015)MathSciNetCrossRefMATHGoogle Scholar - 11.Ecker, K., Huisken, G.: Mean curvature evolution of entire graphs. Ann. Math.
**130**, 453–471 (1989)MathSciNetCrossRefMATHGoogle Scholar - 12.Guan, P., Li, J.: A mean curvature type flow in space forms. Int. Math. Res. Not.
**2015**, 4716–4740 (2015)MathSciNetCrossRefMATHGoogle Scholar - 13.Huisken, G.: Flow by mean curvature of convex surfaces into spheres. J. Differ. Geom.
**22**, 237–266 (1984)MathSciNetCrossRefMATHGoogle Scholar - 14.Huisken, G.: The volume preserving mean curvature flow. J. Reine Angew. Math.
**382**, 35–48 (1987)MathSciNetMATHGoogle Scholar - 15.Huisken, G.: Asymptotic behavior for singularities of the mean curvature flow. J. Differ. Geom.
**31**, 285–299 (1990)MathSciNetCrossRefMATHGoogle Scholar - 16.Huisken, G.: Local and global behaviour of hypersurfaces moving by mean curvature, Differential geometry: partial differential equations on manifolds (Los Angeles, CA, 1990), 175C191, Proceedings of Symposium on Pure Mathematics, 54, Part 1, American Mathematical Society, Providence, RI (1993)Google Scholar
- 17.Kapouleas, N., Kleene, S.J., Møller, N.M.: Mean curvature self-shrinkers of high genus: non-compact examples. J. Reine Angew. Math. (2015). https://doi.org/10.1515/crelle-2015-0050 Google Scholar
- 18.Kleene, S., Møller, N.M.: Self-shrinkers with a rotation symmetry. Trans. Am. Math. Soc.
**366**, 3943–3963 (2014)CrossRefMATHGoogle Scholar - 19.Lawson, H.B.: Local rigidity theorem for minimal surfaces. Ann. Math.
**89**, 187–197 (1969)MathSciNetCrossRefMATHGoogle Scholar - 20.Le, N.Q., Sesum, N.: Blow-up rate of the mean curvature during the mean curvature flow and a gap theorem for self-shrinkers. Commun. Anal. Geom.
**19**, 1–27 (2011)MathSciNetCrossRefMATHGoogle Scholar - 21.Li, H., Wei, Y.: Classification and rigidity of self-shrinkers in the mean curvature flow. J. Math. Soc. Jpn.
**66**, 709–734 (2014)MathSciNetCrossRefMATHGoogle Scholar - 23.Møller, N.M.: Closed self-shrinking surfaces in \({\mathbb{R}}^3\) via the torus. arXiv:1111.7318
- 24.Nguyen, X.H.: Construction of complete embedded self-similar surfaces under mean curvature flow. Part I. Trans. Am. Math. Soc.
**361**, 1683–1701 (2009)CrossRefMATHGoogle Scholar - 25.Nguyen, X.H.: Construction of complete embedded self-similar surfaces under mean curvature flow. Part II. Adv. Differ. Equ.
**15**, 503–530 (2010)MATHGoogle Scholar - 25.Nguyen, X.H.: Construction of complete embedded self-similar surfaces under mean curvature flow. Part III. Duke Math. J.
**163**, 2023–2056 (2014)MathSciNetCrossRefMATHGoogle Scholar - 27.Schoen, R.M., Simon, L.M., Yau, S.T.: Curvature estimates for minimal hypersurfaces. Acta Math.
**134**, 275–288 (1975)MathSciNetCrossRefMATHGoogle Scholar - 28.White, B.: Stratification of minimal surfaces, mean curvature flows, and harmonic maps. J. Reine Angew. Math.
**488**, 1–35 (1997)MathSciNetMATHGoogle Scholar