We show that any Yamabe flow starting from any Riemannian manifold of dimension $m\geq3$ minus a closed submanifold of dimension $n<\frac{m-2}{2}$ must remain geodesically incomplete as long as the flow exists. This is contrasted with the two-dimensional case, where instantaneously complete Yamabe flows always exist. Moreover, we prove that the removability of $n$-dimensional singularities is preserved along the Yamabe flow on closed manifolds of dimension $m\geq3$ if $n<\frac{m-2}{2}$. In the case that $m$ is odd we show that the condition on $n$ is sharp.

中文翻译：

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不完全 Yamabe 流和可移除奇点

我们表明，只要流动存在，任何从维度为 $m\geq3$ 的黎曼流形减去维度为 $n<\frac{m-2}{2}$ 的闭合子流形的任何 Yamabe 流都必须保持测地不完整。这与二维情况形成对比，在二维情况下，瞬时完整的 Yamabe 流始终存在。此外，我们证明，如果 $n<\frac{m-2}{2}$ 维数为 $m\geq3$ 的闭合流形上的 Yamabe 流沿保持 $n$ 维奇点的可移除性。在 $m$ 是奇数的情况下，我们表明 $n$ 上的条件是尖锐的。