Abstract
For a compact spin Riemannian manifold (M,gTM) of dimension n such that the associated scalar curvature kTM verifies that kTM ⩾ n(n − 1), Llarull’s rigidity theorem says that any area-decreasing smooth map f from M to the unit sphere \(\mathbb{S}^{n}\) of nonzero degree is an isometry. We present in this paper a new proof of Llarull’s rigidity theorem in odd dimensions via a spectral flow argument. This approach also works for a generalization of Llarull’s theorem when the sphere \(\mathbb{S}^{n}\) is replaced by an arbitrary smooth strictly convex closed hypersurface in ℝn+1. The results answer two questions by Gromov (2023).
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Acknowledgements
Yihan Li was supported by Nankai Zhide Foundation. Guangxiang Su was supported by National Natural Science Foundation of China (Grant Nos. 12271266 and 11931007), Nankai Zhide Foundation, and the Fundamental Research Funds for the Central Universities (Grant No. 100-63233103). Xiangsheng Wang was supported by National Natural Science Foundation of China (Grant No. 12101361), the Project of Young Scholars of Shandong University, and the Fundamental Research Funds of Shandong University (Grant No. 2020GN063). The authors thank Professor Weiping Zhang for helpful discussions and thank the referees for careful reading and suggestions.
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Li, Y., Su, G. & Wang, X. Spectral flow, Llarull’s rigidity theorem in odd dimensions and its generalization. Sci. China Math. 67, 1103–1114 (2024). https://doi.org/10.1007/s11425-023-2138-5
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DOI: https://doi.org/10.1007/s11425-023-2138-5