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Positive Ricci curvature through Cheeger deformations

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Abstract

This paper is devoted to a deep analysis of the process known as Cheeger deformation, applied to manifolds with isometric group actions. Here, we provide new curvature estimates near singular orbits and present several applications. As the main result, we answer a question raised by a seminal result of Searle–Wilhelm about lifting positive Ricci curvature from the quotient of an isometric action. To answer this question, we develop techniques that can be used to provide a substantially streamlined version of a classical result of Lawson and Yau, generalize a curvature condition of Chavéz, Derdzinski, and Rigas, as well as, give an alternative proof of a result of Grove and Ziller.

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Notes

  1. C. Searle and F. Wilhelm were already aware that Cheeger deformation yields this result.

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Acknowledgements

The authors thank the anonymous referee for criticism, comments, and suggestions, which substantially improved the exposition. They also thank M. Alexandrino, L. Gomes, M. Mazatto, and D. Fadel for helpful discussions on very early versions of this paper. The third-named author expresses gratitude to C. Searle and F. Wilhelm for bringing this problem to his attention, as well as the hospitality of Universität zu Köln. This work was supported by Fundação de Amparo à Pesquisa do Estado de São Paulo [2017/24680-1 to L.C., 2017/10892-7 and 2017/19657-0 to L.S.]; Coordenação de Aperfeiçoamento de Pessoal de Nível Superior [88882.329041/2019-01 to R.S and PROEX to L.C]; Conselho Nacional de Pesquisa [404266/2016-9 to L.S.] and the SNSF-Project 200020E_193062 and the DFG-Priority programme SPP 2026 to L.C during his postdoc position at the University of Fribourg. L.C also thanks the liberty and advices given by Prof. Anand Dessai during that former postdoc position.

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Authors and Affiliations

  1. Instituto de Matemática, Estatística e Computação Cinetífica – Unicamp, Rua Sérgio Buarque de Holanda, 651, Campinas, SP, 13083-859, Brazil

    Leonardo F. Cavenaghi

  2. Instituto de Matemática, Estatística e Computação Científica – UNICAMP, Rua Sérgio Buarque de Holanda, 651, Campinas, SP, 13083-97, Brazil

    Renato J. M. e Silva

  3. Instituto de Ciência e Tecnologia – Unifesp, Avenida Cesare Mansueto Giulio Lattes, 1201, São José dos Campos, SP, 12247-014, Brazil

    Llohann D. Sperança

Authors
  1. Leonardo F. Cavenaghi
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  2. Renato J. M. e Silva
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  3. Llohann D. Sperança
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Correspondence to Leonardo F. Cavenaghi.

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Cavenaghi, L.F., e Silva, R.J.M. & Sperança, L.D. Positive Ricci curvature through Cheeger deformations. Collect. Math. 75, 481–510 (2024). https://doi.org/10.1007/s13348-023-00396-7

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  • DOI: https://doi.org/10.1007/s13348-023-00396-7

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