Examples of topological sphere $λ$-self-expander

Abstract

A $λ$-self-expander $F: M^n\to \mathbb{R}^{n+1}$ is the solution of the isoperimetric problem of weight $e^{\frac{|x|^2}{4}}$. In this paper, for sufficiently large $λ$, we construct an immersed non-embedded spherical $λ$-self-expander in $\mathbb{R}^{n+1}$ with the mean curvature $H>0$. Moreover, we obtain the explicit solution for $n=2$.