Existence and properties of ancient solutions of the Yamabe flow

Shu-Yu Hsu

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arXiv · 1606.02914

Existence and properties of ancient solutions of the Yamabe flow

Shu-Yu Hsu

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Abstract

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.

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Shu-Yu Hsu. 2016-06-11. Existence and properties of ancient solutions of the Yamabe flow. https://arxiv.org/abs/1606.02914

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