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arXiv · 1907.02059

Incomplete Yamabe flows and removable singularities

Mario B. Schulz
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Abstract

We study the Yamabe flow on a Riemannian manifold of dimension $m\geq3$ minus a closed submanifold of dimension $n$ and prove that there exists an instantaneously complete solution if and only if $n>\frac{m-2}{2}$. In the remaining cases $0\leq n\leq\frac{m-2}{2}$ including the borderline case, we show that the removability of the $n$-dimensional singularity is necessarily preserved along the Yamabe flow. In particular, the flow must remain geodesically incomplete as long as it exists. This is contrasted with the two-dimensional case, where instantaneously complete solutions always exist.

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Mario B. Schulz. 2019-07-03. Incomplete Yamabe flows and removable singularities. https://doi.org/10.1016/j.jfa.2020.108475

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