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arXiv · 2008.07028

Rigidity results for complete manifolds with nonnegative scalar curvature

Jintian Zhu
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Abstract

In this paper, we are going to show some rigidity results for complete open Riemannian manifolds with nonnegative scalar curvature. Without using the famous Cheeger-Gromoll splitting theorem we give a new proof to a rigidity result for complete manifolds with nonnegative scalar curvature admitting a proper smooth map to $T^{n-1}\times \mathbf R$ with nonzero degree. Here we introduce a trick to obtain the compactness of limit hypersurface from locally graphical convergence. Based on the same idea we also establish an optimal $2$-systole inequality for several classes of complete Riemannian manifolds with positive scalar curvature and further prove a rigidity result for the equality case.

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Jintian Zhu. 2020-08-16. Rigidity results for complete manifolds with nonnegative scalar curvature. https://arxiv.org/abs/2008.07028

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