Existence of hypercylinder expanders of the inverse mean curvature flow

Abstract

We will give a new proof of the existence of hypercylinder expander of the inverse mean curvature flow which is a radially symmetric homothetic soliton of the inverse mean curvature flow in ${\mathbb{R}}^n\times \mathbb{R}$, $n\ge 2$, of the form $(r,y(r))$ or $(r(y),y)$ where $r=|x|$, $x\in {\mathbb{R}}^n$, is the radially symmetric coordinate and $y\in \mathbb{R}$. More precisely for any $\lambda >\frac{1}{n-1}$ and $\mu >0$, we will give a new proof of the existence of a unique even solution $r(y)$ of the equation $\frac{r^{\prime \prime }(y)}{1+r^{\prime }(y{)}^2}=\frac{n-1}{r(y)}-\frac{1+r^{\prime }(y{)}^2}{\lambda (r(y)-y{r}^{\prime }(y))}$ in $\mathbb{R}$ which satisfies $r(0)=\mu$, $r^{\prime }(0)=0$ and $r(y)>y{r}^{\prime }(y)>0$ for any $y\in \mathbb{R}$. We will prove that ${\lim }_{y\to \infty }r(y)=\infty$ and $a_1:={\lim }_{y\to \infty }{r}^{\prime }(y)$ exists with $0\le a_1<\infty$. We will also give a new proof of the existence of a constant $y_1>0$ such that $r^{\prime \prime }(y_1)=0$, $r^{\prime \prime }(y)>0$ for any $0<y<y_1$ and $r^{\prime \prime }(y)<0$ for any $y>y_1$.

arXiv comment12 pages, some proofs changed, reference updated