# Incomplete Yamabe flows and removable singularities

## TL;DR

This paper investigates the Yamabe flow on manifolds with singularities, establishing conditions for instantaneously complete solutions and the preservation of singularity removability, highlighting differences between higher and two-dimensional cases.

## Contribution

It provides a characterization of when instantaneously complete solutions exist for Yamabe flow on manifolds with singularities and shows the invariance of singularity removability in higher dimensions.

## Key findings

- Existence of instantaneously complete solutions depends on the dimension of the singularity.
- Removability of singularities is preserved along the flow in certain cases.
- Flow remains incomplete if the singularity is not removable.

## 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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Source: https://tomesphere.com/paper/1907.02059