Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 16 (2020), 033, 7 pages      arXiv:1912.03649
Contribution to the Special Issue on Scalar and Ricci Curvature in honor of Misha Gromov on his 75th Birthday

Nonnegative Scalar Curvature and Area Decreasing Maps

Weiping Zhang
Chern Institute of Mathematics & LPMC, Nankai University, Tianjin 300071, P.R. China

Received December 18, 2019, in final form April 15, 2020; Published online April 22, 2020

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.

Key words: scalar curvature; spin manifold; area decreasing map.

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