Scalar curvature and the multiconformal class of a direct product Riemannian manifold

TL;DR

This paper characterizes when a multiconformal class of a product manifold admits positive scalar curvature metrics, linking it to the conformal classes of individual factors, and explores the existence of negative scalar curvature metrics with large volume.

Contribution

It establishes a precise criterion for positive scalar curvature in multiconformal classes based on factors' conformal classes and constructs negative scalar curvature metrics with arbitrarily large volume.

Findings

Positive scalar curvature in multiconformal class iff some factor has a conformal class with positive scalar curvature

Existence of negative scalar curvature metrics with arbitrarily large volume within the multiconformal class

Negative scalar curvature metrics of non-warped product type for certain cases

Abstract

For a closed, connected direct product Riemannian manifold $(M, g)=(M_1\times\cdots\times M_l, g_1\oplus\cdots\oplus g_l)$, we define its multiconformal class $ [\![ g ]\!]$ as the totality $\{f_1^2g_1\oplus \cdots\oplus f_l^2g_l\}$ of all Riemannian metrics obtained from multiplying the metric $g_i$ of each factor $M_i$ by a function $f_i^2>0$ on the total space $M$. A multiconformal class $ [\![ g ]\!]$ contains not only all warped product type deformations of $g$ but also the whole conformal class $[\tilde{g}]$ of every $\tilde{g}\in [\![ g ]\!]$. In this article, we prove that $ [\![ g ]\!]$ carries a metric of positive scalar curvature if and only if the conformal class of some factor $(M_i, g_i)$ does, under the technical assumption $\dim M_i\ge 2$. We also show that, even in the case where every factor $(M_i, g_i)$ has positive scalar curvature, $ [\![ g ]\!]$ carries a metric of scalar curvature constantly equal to $-1$ and with arbitrarily large volume, provided $l\ge 2$ and $\dim M\ge 3$. In this case, such negative scalar curvature metrics within $ [\![ g ]\!]$ for $l=2$ cannot be of any warped product type.