Existence of self-similar solution of the inverse mean curvature flow

TL;DR

This paper provides a new proof for the existence and uniqueness of radially symmetric self-similar solutions to the inverse mean curvature flow, characterizing their properties and asymptotic behavior.

Contribution

It introduces a novel proof for the existence of self-similar solutions of the inverse mean curvature flow with specific symmetry and growth conditions, extending previous results.

Findings

Existence of a unique radially symmetric solution with specified properties.

Solutions satisfy positivity and convexity conditions.

Asymptotic ratio of radial derivative to the function approaches a constant.

Abstract

We will give a new proof of a recent result of P.~Daskalopoulos, G.Huisken and J.R.King ([DH] and reference [7] of [DH]) on the existence of self-similar solution of the inverse mean curvature flow which is the graph of a radially symmetric solution in $\mathbb{R}^n$, $n\ge 2$, of the form $u(x,t)=e^{\lambda t}f(e^{-\lambda t} x)$ for any constants $\lambda>\frac{1}{n-1}$ and $\mu<0$ such that $f(0)=\mu$. More precisely we will give a new proof of the existence of a unique radially symmetric solution $f$ of the equation $\mbox{div}\,\left(\frac{\nabla f}{\sqrt{1+|\nabla f|^2}} \right)=\frac{1}{\lambda}\cdot\frac{\sqrt{1+|\nabla f|^2}}{x\cdot\nabla f-f}$ in $\mathbb{R}^n$, $f(0)=\mu$, for any $\lambda>\frac{1}{n-1}$ and $\mu<0$, which satisfies $f_r(r)>0$, $f_{rr}(r)>0$ and $rf_r(r)>f(r)$ for all $r>0$. We will also prove that $\lim_{r\to\infty}\frac{rf_r(r)}{f(r)}=\frac{\lambda (n-1)}{\lambda (n-1)-1}$.