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

Abstract

We examine the solutions of 1-dimensional λ-self shrinkers and show that for certain λ < 0, there are some closed, embedded solutions other than circles. For negative λ near zero, there are embedded solutions with 2-symmetry. For negative λ with large absolute value, there are embedded solutions with m-symmetry, where m is greater than 2.Date: September 24, 2018.

Surfaces with a canonical principal direction and prescribed mean curvature

“…Although in the literature, Eqs. (2) and (3) have been studied when μ = 0, the case μ = 0 makes sense in the context of manifolds with density [10,16]. Indeed, consider R 3 with a positive smooth density function e φ , φ ∈ C ∞ (R 3 ), which serves as a weight for the volume and the surface area.…”

“…Then, it is immediate that M is a critical point of A φ for a given weighted volume if and only if H φ = μ is a constant function. Equations (2) and (3) appear with suitable choices of the density function φ; namely, for Eq. (2), we take φ( p) = 2λ p, v , and for Eq.…”

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

“…From [7], Chang has proved there exist a lot of complete embedded λ-curves Γ in R 2 . Hence we have Example 2.3.…”

“…For 1-dimensional self-shrinker in R 2 , Abresch and Langer [1] proved the circle is the only compact embedded self-shrinker. But for λ-curve in R 2 , Chang [7] has proved, for λ < 0, there are many compact embedded λ-curves other than the circle. From the above examples, we know that there are a lot of examples of complete embedded λ-hypersurfaces, which are not self-shrinkers, in R n+1 .…”

A Survey of Closed Self-Shrinkers with Symmetry

“…Recently, classification results and conservation laws for self-shrinkers were developed in other geometric contexts by Halldorsson [28,29] and Chang [13].…”