Classification of singularities in the complete conformally flat Yamabe flow

Details Classification of singularities in the complete conformally flat Yamabe flow Classification of singularities in the complete conformally flat Yamabe flow

2007-05-24

arxiv.org/abs/0705.3667v4

Mathematics

Differential Geometry

The paper has been withdrawn due to a crucial error in the argument

Scientific paper

We show that an eternal solution to a complete, locally conformally flat Yamabe flow, $\frac{\partial}{\partial t} g = -Rg$, with uniformly bounded scalar curvature and positive Ricci curvature at $t = 0$, where the scalar curvature assumes its maximum is a gradient steady soliton. As an application of that, we study the blow up behavior of $g(t)$ at the maximal time of existence, $T < \infty$. We assume that $(M,g(\cdot, t))$ satisfies (i) the injectivity radius bound {\bf or} (ii) the Schouten tensor is positive at time $t = 0$ and the scalar curvature bounded at each time-slice. We show that the singularity the flow develops at time $T$ is always of type I.

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