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.

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.