Infinitely many solutions to the Yamabe problem on noncompact manifolds
Infinitely many solutions to the Yamabe problem on noncompact manifolds
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非紧流形上 Yamabe 问题的无穷多个解
DOI：
10.5802/aif.3172
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发表时间：
2016
期刊：
影响因子：
--
通讯作者：
P. Piccione
中科院分区：
文献类型：
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作者：
R. G. Bettiol;P. Piccione
我们在某些非紧积流形上的规定共形类上建立了无限多个具有恒定标量曲率的完整度量的存在。这些包括具有恒定正标量曲率的闭流形的乘积和非紧或欧几里得类型的简单连接对称空间；特别是，$\mathbb S^m \times\mathbb R^d$、$m\geq2$、$d\geq1$ 和 $\mathbb S^m\times\mathbb H^d$、$2\leq d<m$。因此，我们在 $\mathbb S^m\setminus\mathbb S^k$ 上获得奇异 Yamabe 问题的无限多个周期解，对于所有 $0\leq k<(m-2)/2$（可能存在非唯一性的最大范围）。我们还表明，$Iso(\mathbb R^d)$ 中的所有比伯巴赫群都是 $\mathbb S^m\times\mathbb R^d$、$m\geq2$、$d\geq1$ 上 Yamabe 问题解的分叉分支的周期。
We establish the existence of infinitely many complete metrics with constant scalar curvature on prescribed conformal classes on certain noncompact product manifolds. These include products of closed manifolds with constant positive scalar curvature and simply-connected symmetric spaces of noncompact or Euclidean type; in particular, $\mathbb S^m \times\mathbb R^d$, $m\geq2$, $d\geq1$, and $\mathbb S^m\times\mathbb H^d$, $2\leq d<m$. As a consequence, we obtain infinitely many periodic solutions to the singular Yamabe problem on $\mathbb S^m\setminus\mathbb S^k$, for all $0\leq k<(m-2)/2$, the maximal range where nonuniqueness is possible. We also show that all Bieberbach groups in $Iso(\mathbb R^d)$ are periods of bifurcating branches of solutions to the Yamabe problem on $\mathbb S^m\times\mathbb R^d$, $m\geq2$, $d\geq1$.