Existence of hypercylinder expanders of the inverse mean curvature flow

Hui, Kin Ming

Canadian mathematical bulletin, Tome 65 (2022) no. 3, pp. 543-551

Résumé

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 $\frac {r^{\prime \prime }(y)}{1+r^{\prime }(y)^{2}}=\frac {n-1}{r(y)}-\frac {1+r^{\prime }(y)^{2}}{\lambda (r(y)-yr^{\prime }(y))}$ in $\mathbb {R}$ which satisfies $r(0)=\mu $, $r^{\prime }(0)=0$ and $r(y)>yr^{\prime }(y)>0$ 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 $y>y_{1}$.

Détail

DOI : 10.4153/S0008439521000485

Mots-clés : inverse mean curvature flow, hypercylinder expander solution, existence, asymptotic behavior