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Four-dimensional complete gradient shrinking Ricci solitons with half positive isotropic curvature

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Abstract

In this paper, we investigate the geometry of 4-dimensional complete gradient shrinking Ricci solitons with half positive isotropic curvature (half PIC) or half nonnegative isotropic curvature. Our first main result is a certain form of curvature estimates for such Ricci shrinkers, including a quadratic curvature lower bound estimate for noncompact ones with half PIC. As a consequence, we obtain a new and more direct proof of the classification result, first observed by Li et al. (Int Math Res Not 3:949–959, 2018), for gradient shrinking Kähler–Ricci solitons of complex dimension two with nonnegative isotropic curvature. Moreover, based on a strong maximum principle argument, we classify 4-dimensional complete gradient shrinking Ricci solitons with half nonnegative isotropic curvature (except the half PIC case). Finally, we treat the half PIC case under an additional assumption on the Ricci tensor.

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Data Availability Statement:

Data sharing not applicable to this article as no data sets were generated or analysed during the current study.

Notes

  1. The classifications in dimension \(n\ge 5\) follow from the work of Böhm and Wilking [1].

  2. Later, this was proved in all dimensions \(n\ge 5\) by Brendle and Schoen [2] and Nguyen [42] independently, and this property played an essential role in Brendle-Schoen’s proof of the long standing 1/4-pinching differentiable sphere theorem.

  3. See also [49] for a different proof.

  4. More precisely, the operator \(B:\wedge ^{-}(M)\rightarrow \wedge ^{+}(M)\) is given by \(\mathring{Rc}\bigcirc \!\!\!\!\!\!\wedge \,g\), the Kulkarni-Nomizu product of \(\mathring{Rc}\) and g. In particular, B is identically zero when \((M^4, g)\) is Einstein.

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Acknowledgements

We would like to thank Professor Richard Hamilton and Professor Lei Ni for their interests in our work. We also like to thank Dr. Jiangtao Yu for discussions and the anonymous referee for helpful comments.

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  1. Department of Mathematics, Lehigh University, Bethlehem, PA, 18015, USA

    Huai-Dong Cao & Junming Xie

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  1. Huai-Dong Cao
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Correspondence to Junming Xie.

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Huai-Dong Cao research partially supported by Simons Foundation Collaboration Grant #586694.

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Cao, HD., Xie, J. Four-dimensional complete gradient shrinking Ricci solitons with half positive isotropic curvature. Math. Z. 305, 25 (2023). https://doi.org/10.1007/s00209-023-03356-w

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