Enlargeable metrics on nonspin manifolds
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- by Simone Cecchini and Thomas Schick PDF
- Proc. Amer. Math. Soc. 149 (2021), 2199-2211 Request permission
Abstract:
We show that an enlargeable Riemannian metric on a (possibly nonspin) manifold cannot have uniformly positive scalar curvature. This extends a well-known result of Gromov and Lawson to the nonspin setting. We also prove that every noncompact manifold admits a nonenlargeable metric. In proving the first result, we use the main result of the recent paper by Schoen and Yau on minimal hypersurfaces to obstruct positive scalar curvature in arbitrary dimensions. More concretely, we use this to study nonzero degree maps $f\colon X^n\rightarrow S^k\times T^{n-k}$, with $k=1,2,3$. When $X$ is a closed oriented manifold endowed with a metric $g$ of positive scalar curvature and the map $f$ is (possibly area) contracting, we prove inequalities relating the lower bound of the scalar curvature of $g$ and the contracting factor of the map $f$.References
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Additional Information
- Simone Cecchini
- Affiliation: Mathematisches Institut, Georg-August-Universität, Göttingen, Germany
- MR Author ID: 1194724
- Email: cecchini@mathematik.uni-goettingen.de
- Thomas Schick
- Affiliation: Mathematisches Institut, Georg-August-Universität, Göttingen, Germany
- MR Author ID: 635784
- Email: thomas.schick@math.uni-goettingen.de
- Received by editor(s): October 22, 2018
- Received by editor(s) in revised form: May 4, 2019
- Published electronically: February 24, 2021
- Additional Notes: Both authors thank the DFG SPP 2026 for support
- Communicated by: Jia-Ping Wang
- © Copyright 2021 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 149 (2021), 2199-2211
- MSC (2020): Primary 53C23; Secondary 49Q05
- DOI: https://doi.org/10.1090/proc/14706
- MathSciNet review: 4232210