Shu-Yu Hsu

We give a new proof for the existence of rotationally symmetric steady and expanding gradient Ricci solitons in dimension $n+1$, $2\le n\le 4$, with metric $g=\frac{d{a}^2}{h(a^2)}+a^{2}d\sigma$ for some function $h$ where $d\sigma$ is the standard metric on the unit sphere $S^n$ in ${\mathbb{R}}^n$. More precisely for any $\lambda \ge 0$, $2\le n\le 4$ and ${\mu }_1\in \mathbb{R}$, we prove the existence of unique solution $h\in C^2((0,\infty ))\cap C^1([0,\infty ))$ for the equation $2r^{2}h(r)h_{rr}(r)=(n-1)h(r)(h(r)-1)+r{h}_r(r)(r{h}_r(r)-\lambda r-(n-1))$, $h(r)>0$, in $(0,\infty )$ satisfying $h(0)=1$, $h_r(0)={\mu }_1$. We also prove the existence of unique analytic solution of the about equation on $[0,\infty )$ for any $\lambda \ge 0$, $n\ge 2$ and ${\mu }_1\in \mathbb{R}$. Moreover we will prove the asymptotic behaviour of the solution $h$ for any $n\ge 2$, $\lambda \ge 0$ and ${\mu }_1\in \mathbb{R}\setminus \{0\}$.

We will give a new proof of the existence of non-compact homothetic solitons of the inverse mean curvature flow (cf.) 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 $\frac{r_{yy}(y)}{1+r_y(y{)}^2}=\frac{n-1}{r(y)}-\frac{1+r_y(y{)}^2}{\lambda (r(y)-y{r}_y(y))}$, $r(y)>0$, 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 ${\lim }_{y\to +\infty }y{r}_y(y)=0$.

By using the De Giorgi iteration method we will give a new simple proof of the recent result of B.Kotschwar, O.Munteanu, J.Wang [KMW] and N.Sesum [S] on the local boundedness of the Riemmanian curvature tensor of solutions of Ricci flow in terms of its inital value on a given ball and a local uniform bound on the Ricci curvature.

Let $n\ge 3$ and $m=\frac{n-2}{n+2}$. We construct $5$-parameters, $4$-parameters, $3$-parameters ancient solutions of the equation $v_t=(v^m{)}_{xx}+v-v^m$, $v>0$, in $\mathbb{R}\times (-\infty ,T)$ for some $T\in \mathbb{R}$. This equation arises in the study of Yamabe flow. We obtain various properties of the ancient solutions of this equation including exact decay rate of ancient solutions as $|x|\to \infty$. We also prove that both the $3$-parameters ancient solution and the $4$-parameters ancient solution are singular limit solution of the $5$-parameters ancient solutions.