Another proof of the existence of homothetic solitons of the inverse mean curvature flow

Abstract

We will give a new proof of the existence of non-compact homothetic solitons of the inverse mean curvature flow (cf. \cite{DLW}) 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 $\frac{1}{n}<\lambda<\frac{1}{n-1}$ and $\mu<0$, we will give a new proof of the existence of a unique solution $r(y)\in C^2(\mu,\infty)\cap C([\mu,\infty))$ of the equation $\frac{r_{yy}(y)}{1+r_y(y)^2}=\frac{n-1}{r(y)}-\frac{1+r_y(y)^2}{\lambda(r(y)-yr_y(y))}$, $r(y)>0$, in $(\mu,\infty)$ which satisfies $r(\mu)=0$ and $r_y(\mu)=\lim_{y\searrow\mu}r_y(y)=+\infty$. We also prove that there exist constants $y_2>y_1>0$ such that $r_y(y)>0$ for any $\mu<y<y_1$, $r_y(y_1)=0$, $r_y(y)<0$ for any $y>y_1$, $r_{yy}(y)<0$ for any $\mu<y<y_2$, $r_{yy}(y_2)=0$ and $r_{yy}(y)>0$ for any $y>y_2$. Moreover $\lim_{y\to +\infty}r(y)=0$ and $\lim_{y\to +\infty}yr_y(y)=0$.