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

Abstract

In this paper, we introduce a definition of λ-hypersurfaces of weighted volume-preserving mean curvature flow in Euclidean space. We prove that λhypersurfaces are critical points of the weighted area functional for the weighted volume-preserving variations. Furthermore, we classify complete λ-hypersurfaces with polynomial area growth and H−λ ≥ 0, which are generalizations of the results due to Huisken [19], . We also define a F -functional and study F -stability of λ-hypersurfaces, which extend a result of Colding-Minicozzi [11]. Lower bound growth and upper bound growth of the area for complete and non-compact λ-hypersurfaces are also studied.2001 Mathematics Subject Classification: 53C44, 53C42. Key words and phrases: the weighted volume-preserving mean curvature flow, the weighted area functional, F -stability, weak stability, λ-hypersurfaces.

Variational characterizations of $$\xi $$-submanifolds in the Eulicdean space $${{\mathbb {R}}}^{m+p}$$

“…In Mar., 2014, Cheng and Wei formally introduced ( [9], finally revised in May, 2015) the definition of λ-hypersurface of weighted volume-preserving mean curvature flow in Euclidean space, giving a natural generalization of self-shrinkers in the hypersurface case. According to [9], a hypersurface x : M m → R m+1 is called a λ-hypersurface if its (scalar-valued) mean curvature H satisfies…”

A new pinching theorem for complete self-shrinkers and its generalization

“…More generally, we consider the rigidity of λ-hypersurfaces. The concept of λhypersurfaces was introduced independently by Cheng-Wei [7] via the weighted volume-preserving mean curvature flow and McGonagle-Ross [25] via isoperimetric type problem in a Gaussian weighted Euclidean space. Precisely, the hypersurfaces of Euclidean space satisfying the following equation are called λ-hypersurfaces:…”

“…where X N is the projection of X on the unit normal vector ξ and λ is a constant. In recent years, the rigidity of λ-hypersurfaces has been investigated by several authors [5,7,15,33]. In [15], Guang showed that if M is a λ-hypersurface with polynomial volume growth in R n+1 , and if |A| 2 ≤ α λ , then M must be one of the generalized cylinders, where α λ = 1 2 (2 + λ 2 − |λ| √ λ 2 + 4).…”

1-Dimensional solutions of the $$\lambda $$ λ -self shrinkers

“…where H is the mean curvature, N is the normal vector on the surface and λ is a constant. This equation is first studied by McGonagle and Ross [14] and be named as λ-hypersurface in the work of Cheng and Wei [7]. The equation arises in the Gaussian isoperimetric problem: In R n+1 , the weighted Gaussian volume element and area element are given by dV µ = exp(− |x| 2 4 )dV and dA µ = exp(− |x| 2 4 )dA, where dV and dA are the volume element and area element induced by the Euclidean metric.…”

“…The λ-hypersurfaces also arise in the study of the weighted volume-preserving flow by Cheng and Wei [7]. Note that the "weighted volume" in their work is defined on the surface and is different from the Gaussian weighted volume above.…”