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Existence of singular rotationally symmetric gradient Ricci solitons in higher dimensions
- Part of
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- Journal:
- Canadian Mathematical Bulletin / Volume 67 / Issue 3 / September 2024
- Published online by Cambridge University Press:
- 21 March 2024, pp. 842-859
- Print publication:
- September 2024
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- Article
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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 
$\lambda \ge 0$ and 
$c_0>0$, we prove that there exist infinitely many solutions 
${h\in C^2((0,\infty );\mathbb {R}^+)}$ for the equation 
$2r^2h(r)h_{rr}(r)=(n-1)h(r)(h(r)-1)+rh_r(r)(rh_r(r)-\lambda r-(n-1))$, 
$h(r)>0$, in 
$(0,\infty )$ satisfying 
$\underset {\substack {r\to 0}}{\lim }\,r^{\sqrt {n}-1}h(r)=c_0$ and prove the higher-order asymptotic behavior 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 behavior near the origin.
