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

Abstract

In this paper, we introduce a definition of $\lambda$-hypersurfaces of weighted volume-preserving mean curvature flow in 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 $H-\lambda \ge 0$, which are generalizations of the results due to Huisken, Colding-Minicozzi. We also define a $\mathcal{F}$-functional and study $\mathcal{F}$-stability of $\lambda$-hypersurfaces, which extend a result of Colding-Minicozzi. Lower bound growth and upper bound growth of the area for complete and non-compact $\lambda$-hypersurfaces are also studied.

46 pages

References in corpus (3)

• Classification and rigidity of self-shrinkers in the Mean curvature flow
• A mean curvature type flow in space forms
• One dimensional solutions of the $\lambda$-self shrinkers

Cited by in corpus (7)

• A new pinching theorem for complete self-shrinkers and its generalization
• Compact embedded $\lambda$-torus in Euclidean spaces
• A global pinching theorem of complete $\lambda$-hypersurfaces
• One dimensional solutions of the $\lambda$-self shrinkers
• Complete $\lambda$-surfaces in ${\mathbb{R}}^3$
• Volume growth of complete submanifolds in gradient Ricci Solitons with bounded weighted mean curvature
• A remark on a Bernstein type result for $\lambda$-hypersurfaces