No metrics with Positive Scalar Curvatures on Aspherical 5-Manifolds

Abstract

A metric space $X$ is called uniformly acyclic if there there exists an {\it acyclicty control function} $R=R(r)=R_X(r)\ge r$, $0\le r<\infty$, such that the homology inclusion homomorphisms between the balls around all points $x\in X$, $$H_i(B_x(r))\to H_i(B_x(R))$$ vanish for all $i=1,2,\dots$. We show that if a complete orientable $m$-dimensional manifold $\tilde{X}$ of dimension $m\le 5$ admits a proper (infinity goes to infinity) distance decreasing map to a complete $m$-dimensional uniformly acyclic manifold, then the scalar curvature of $\tilde{X}$ can't be uniformly positive, $$\underset{x\in \tilde{X}}{\inf }Sc(X,x)\le 0.$$ Since the universal coverings $\tilde{X}$ of compact aspherical manifolds $X$ are {\it uniformly acyclic}, (in fact, {\it uniformly contractible}), these $X$, admit no metrics with $Sc>0$ for $dim(X)\le 5$. Our argument, that depends on {\it torical symmetrization} of {\it stable $\mu$-bubbles}, is inspired by the recent paper by Otis Chodosh and Chao Li on non-existence of metrics with $Sc>0$ on aspherical 4-manifolds and is also influenced by the ideas of Jintian Zhu and Thomas Richard.

References in corpus (2)

• Rigidity of Area-Minimizing $2$-Spheres in $n$-Manifolds with Positive Scalar Curvature
• A short proof of Gromov's filling inequality

Cited by in corpus (1)

• Positive Scalar Curvature on Noncompact Manifolds and the Liouville Theorem