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Nonnegative scalar curvature and area decreasing maps on complete foliated manifolds

  • Guangxiang Su EMAIL logo , Xiangsheng Wang and Weiping Zhang

Abstract

Let ( M , g T M ) be a noncompact complete Riemannian manifold of dimension n, and let F T M be an integrable subbundle of TM. Let g F = g T M | F be the restricted metric on F and let k F be the associated leafwise scalar curvature. Let f : M S n ( 1 ) be a smooth area decreasing map along F, which is locally constant near infinity and of non-zero degree. We show that if k F > rk ( F ) ( rk ( F ) - 1 ) on the support of d f , and either TM or F is spin, then inf ( k F ) < 0 . As a consequence, we prove Gromov’s sharp foliated ε -twisting conjecture. Using the same method, we also extend two famous non-existence results due to Gromov and Lawson about Λ 2 -enlargeable metrics (and/or manifolds) to the foliated case.

Funding statement: Guangxiang Su and Weiping Zhang were partially supported by NSFC Grant No. 11931007 and Nankai Zhide Foundation. Xiangsheng Wang was partially supported by NSFC Grant No. 12101361, the project of Young Scholars of SDU and the fundamental research funds of Shandong University, Grant No. 2020GN063.

Acknowledgements

The authors would like to thank the anonymous referee for careful reading and valuable suggestions.

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Received: 2021-07-07
Revised: 2022-05-25
Published Online: 2022-07-23
Published in Print: 2022-09-01

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