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This article is cited in 6 scientific papers (total in 6 papers)
Nonnegative Scalar Curvature and Area Decreasing Maps
Weiping Zhang Chern Institute of Mathematics & LPMC, Nankai University, Tianjin 300071, P.R. China
Abstract:
Let $\big(M,g^{TM}\big)$ be a noncompact complete spin Riemannian manifold of even dimension $n$, with $k^{TM}$ denote the associated scalar curvature. Let $f\colon M\rightarrow S^{n}(1)$ be a smooth area decreasing map, which is locally constant near infinity and of nonzero degree. We show that if $k^{TM}\geq n(n-1)$ on the support of ${\rm d}f$, then $ \inf \big(k^{TM}\big)<0$. This answers a question of Gromov. We use a simple deformation of the Dirac operator to prove the result. The odd dimensional analogue is also presented.
Keywords:
scalar curvature, spin manifold, area decreasing map.
Received: December 18, 2019; in final form April 15, 2020; Published online April 22, 2020
Citation:
Weiping Zhang, “Nonnegative Scalar Curvature and Area Decreasing Maps”, SIGMA, 16 (2020), 033, 7 pp.
Linking options:
https://www.mathnet.ru/eng/sigma1570 https://www.mathnet.ru/eng/sigma/v16/p33
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Abstract page: | 187 | Full-text PDF : | 26 | References: | 21 |
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