Upper bound preservation of the total scalar curvature in a conformal class

TL;DR

This paper investigates the stability of upper bounds on total scalar curvature within conformal classes on closed manifolds, revealing conditions under which these bounds are preserved under limits, especially relating to Yamabe constants.

Contribution

It establishes the $C^{0}$-closedness of upper bounds on total scalar curvature in conformal classes with nonpositive Yamabe constant and characterizes the structure when the Yamabe constant is positive.

Findings

Upper bound condition is $C^{0}$-closed if Yamabe constant is nonpositive.

For positive Yamabe constant, certain metric sets are $C^{0}$-closed.

Provides conditions for stability of scalar curvature bounds in conformal classes.

Abstract

We show that in an arbitrarily fixed conformal class on a closed manifold, the upper bound condition of the total scalar curvature is $C^{0}$-closed if its Yamabe constant is nonpositive. Moreover, we show that if a conformal class on a closed manifold has positive Yamabe constant, then the intersection of such conformal class and the space of all Riemannian metrics, whose scalar curvatures are bounded from below as well as total scalar curvatures are bounded from above is $C^{0}$-closed in the space of all Riemannian metrics.