Self-shrinkers are an important class of solutions to the mean curvature flow and their generalization is λ-hypersurfaces. In this paper, we study λ-hypersurfaces and give a rigidity result about complete λ-hypersurfaces.

A note on rigidity theorem of λ-hypersurfaces

In Euclidean space, we investigate surfaces whose mean curvature H satisfies the equation $$H=\alpha \langle N,{\mathbf {x}}\rangle +\lambda $$
 , where N is the Gauss map, $${\mathbf {x}}$$
 is the position vector, and $$\alpha $$
 and $$\lambda $$
 are two constants. There surfaces generalize self-shrinkers and self-expanders of the mean curvature flow. We classify the ruled surfaces and the translation surfaces, proving that they are cylindrical surfaces.

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Ruled Surfaces of Generalized Self-Similar Solutions of the Mean Curvature Flow

The purpose of this paper is to study complete $\lambda$-surfaces in Euclidean space $\mathbb R^3$. A complete classification for 2-dimensional complete $\lambda$-surfaces in Euclidean space $\mathbb R^3$ with constant squared norm of the second fundamental form is given.

Complete $\lambda$-surfaces in $\mathbb R^3$

The purpose of this paper is to study complete $$\lambda $$
 -surfaces in Euclidean space $${\mathbb {R}}^3$$
 . A complete classification for 2-dimensional complete $$\lambda $$
 -surfaces in Euclidean space $$\mathbb R^3$$
 with constant squared norm of the second fundamental form is given, which confirms a conjecture of Guang (Self-shrinkers and translating solitons of mean curvature flow, 2016, p 74).

Complete λ-Hypersurfaces in Euclidean Spaces

In this paper we show the existence of a closed, embedded $$\lambda $$
 -hypersurfaces $$\Sigma \subset \mathbb {R}^{2n}$$
 . The hypersurface $$\Sigma $$
 is diffeomorphic to $$\mathbb {S}^{n-1} \times \mathbb {S}^{n-1} \times \mathbb {S}^1$$
 and exhibits $$SO(n) \times SO(n)$$
 symmetry. Our approach uses a “shooting method” similar to the approach used by McGrath in constructing a generalized self-shrinking “torus” solution to mean curvature flow. The result generalizes the $$\lambda $$
 torus found by Cheng and Wei.

John Ross

Papers

On the existence of a closed, embedded, rotational $$\lambda $$ λ -hypersurface