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arXiv · 1705.08886

$ε$-Regularity and Structure of 4-dimensional Shrinking Ricci Solitons

Shaosai Huang
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Abstract

A closed four dimensional manifold cannot possess a non-flat Ricci soliton metric with arbitrarily small $L^2$-norm of the curvature. In this paper, we localize this fact in the case of shrinking Ricci solitons by proving an $\varepsilon$-regularity theorem, thus confirming a conjecture of Cheeger-Tian. As applications, we will also derive structural results concerning the degeneration of the metrics on a family of complete non-compact four dimensional shrinking Ricci solitons without a uniform entropy lower bound. In the appendix, we provide a detailed account of the equivariant good chopping theorem when collapsing with locally bounded curvature happens.

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Shaosai Huang. 2018-09-06. $ε$-Regularity and Structure of 4-dimensional Shrinking Ricci Solitons. https://doi.org/10.1093/imrn%2Frny069

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