Existence of singular rotationally symmetric gradient Ricci solitons in higher dimensions
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2021
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| _version_ | 1866912226970959872 |
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| author | Hui, Kin Ming |
| author_facet | Hui, Kin Ming |
| contents | By using fixed point argument we give a proof for the existence of singular rotationally symmetric steady and expanding gradient Ricci solitons in higher dimensions with metric $g=\frac{da^2}{h(a^2)}+a^2g_{S^n}$ for some function $h$ where $g_{S^n}$ is the standard metric on the unit sphere $S^n$ in $\mathbb{R}^n$ for any $n\ge 2$. More precisely for any $λ\ge 0$ and $c_0>0$, we prove that there exist infinitely many solutions $h\in C^2((0,\infty);\mathbb{R}^+)$ for the equation $2r^2h(r)h_{rr}(r)=(n-1)h(r)(h(r)-1)+rh_r(r)(rh_r(r)-λr-(n-1))$, $h(r)>0$, in $(0,\infty)$ satisfying $\underset{\substack{r\to 0}}{\lim}\,r^{\sqrt{n}-1}h(r)=c_0$ and prove the higher order asymptotic behaviour of the global singular solutions near the origin. We also find conditions for the existence of unique global singular solution of such equation in terms of its asymptotic behaviour near the origin. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2107_13685 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Existence of singular rotationally symmetric gradient Ricci solitons in higher dimensions Hui, Kin Ming Differential Geometry Analysis of PDEs Primary 53E20, 35J75 Secondary 53C21 By using fixed point argument we give a proof for the existence of singular rotationally symmetric steady and expanding gradient Ricci solitons in higher dimensions with metric $g=\frac{da^2}{h(a^2)}+a^2g_{S^n}$ for some function $h$ where $g_{S^n}$ is the standard metric on the unit sphere $S^n$ in $\mathbb{R}^n$ for any $n\ge 2$. More precisely for any $λ\ge 0$ and $c_0>0$, we prove that there exist infinitely many solutions $h\in C^2((0,\infty);\mathbb{R}^+)$ for the equation $2r^2h(r)h_{rr}(r)=(n-1)h(r)(h(r)-1)+rh_r(r)(rh_r(r)-λr-(n-1))$, $h(r)>0$, in $(0,\infty)$ satisfying $\underset{\substack{r\to 0}}{\lim}\,r^{\sqrt{n}-1}h(r)=c_0$ and prove the higher order asymptotic behaviour of the global singular solutions near the origin. We also find conditions for the existence of unique global singular solution of such equation in terms of its asymptotic behaviour near the origin. |
| title | Existence of singular rotationally symmetric gradient Ricci solitons in higher dimensions |
| topic | Differential Geometry Analysis of PDEs Primary 53E20, 35J75 Secondary 53C21 |
| url | https://arxiv.org/abs/2107.13685 |