ε-Regularity and Structure of Four-dimensional Shrinking Ricci Solitons
2018 ◽
Vol 2020
(5)
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pp. 1511-1574
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Keyword(s):
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 gradient shrinking Ricci solitons by proving an $\varepsilon $-regularity theorem, thus confirming a conjecture of Cheeger–Tian [20]. As applications, we will also derive structural results concerning the degeneration of the metrics on a family of complete non-compact four-dimensional gradient 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.
2016 ◽
Vol 19
(01)
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pp. 1650001
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2019 ◽
Vol 16
(05)
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pp. 1950073
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2021 ◽
pp. 2150052
Keyword(s):
2019 ◽
Vol 16
(09)
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pp. 1950134
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Keyword(s):
2017 ◽
Vol 55
(2)
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pp. 143-156
Keyword(s):
2014 ◽
Vol 25
(11)
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pp. 1450104
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Keyword(s):