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 in ℝ n × ℝ {\mathbb{R}^{n}\times\mathbb{R}} , n ≥ 2 {n\geq 2} , of the form ( r , y ⁢ ( r ) ) {(r,y(r))} or ( r ⁢ ( y ) , y ) {(r(y),y)} , where r = | x | {r=|x|} , x ∈ ℝ n {x\in\mathbb{R}^{n}} , is the radially symmetric coordinate and y ∈ ℝ {y\in\mathbb{R}} . More precisely for any 1 n < λ < 1 n - 1 {\frac{1}{n}<\lambda<\frac{1}{n-1}} and μ < 0 {\mu<0} , we will give a new proof of the existence of a unique solution r ⁢ ( y ) ∈ C 2 ⁢ ( μ , ∞ ) ∩ C ⁢ ( [ μ , ∞ ) ) {r(y)\in C^{2}(\mu,\infty)\cap C([\mu,\infty))} of the equation r y ⁢ y ⁢ ( y ) 1 + r y ⁢ ( y ) 2 = n - 1 r ⁢ ( y ) - 1 + r y ⁢ ( y ) 2 λ ⁢ ( r ⁢ ( y ) - y ⁢ r y ⁢ ( y ) ) , r ⁢ ( y ) > 0 , \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))},\quad r(y)>0, in ( μ , ∞ ) {(\mu,\infty)} which satisfies r ⁢ ( μ ) = 0 {r(\mu)=0} and r y ⁢ ( μ ) = lim y ↘ μ ⁡ r y ⁢ ( y ) = + ∞ {r_{y}(\mu)=\lim_{y\searrow\mu}r_{y}(y)=+\infty} . We prove that there exist constants y 2 > y 1 > 0 y_{2}>y_{1}>0 such that r y ⁢ ( y ) > 0 r_{y}(y)>0 for any μ < y < y 1 \mu<y<y_{1} , r y ⁢ ( y 1 ) = 0 r_{y}(y_{1})=0 , r y ⁢ ( y ) < 0 r_{y}(y)<0 for any y > y 1 y>y_{1} , r y ⁢ y ⁢ ( y ) < 0 r_{yy}(y)<0 for any μ < y < y 2 \mu<y<y_{2} , r y ⁢ y ⁢ ( y 2 ) = 0 r_{yy}(y_{2})=0 and r y ⁢ y ⁢ ( y ) > 0 {r_{yy}(y)>0} for any y > y 2 {y>y_{2}} . Moreover, lim y → + ∞ ⁡ r ⁢ ( y ) = 0 {\lim_{y\to+\infty}r(y)=0} and lim y → + ∞ ⁡ y ⁢ r y ⁢ ( y ) = 0 {\lim_{y\to+\infty}yr_{y}(y)=0} .