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arXiv · 0705.3667

Classification of singularities in the complete conformally flat Yamabe flow

Panagiota Daskalopoulos·Natasa Sesum
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Abstract

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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Panagiota Daskalopoulos, Natasa Sesum. 2012-03-05. Classification of singularities in the complete conformally flat Yamabe flow. https://arxiv.org/abs/0705.3667

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