Llarull's theorem on odd dimensional manifolds: the noncompact case

Yihan Li; Guangxiang Su; Xiangsheng Wang; Weiping Zhang

paper

Llarull's theorem on odd dimensional manifolds: the noncompact case

arXiv:2404.18153

Abstract

Let $(M, g^{T M})$ be an odd dimensional ( $dim M \geq 3$ ) connected oriented noncompact complete spin Riemannian manifold. Let $k^{T M}$ be the associated scalar curvature. Let $f : M \to S^{d i m M} (1)$ be a smooth area decreasing map which is locally constant near infinity and of nonzero degree. Suppose $k^{T M} \geq (dim M) (dim M - 1)$ on the support of $d f$ , we show that $in f (k^{T M}) < 0$ . This answers a question of Gromov.