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

Abstract

We show that in an arbitrarily fixed conformal class with non-positive Yamabe constant on a closed manifold, the upper bound of the total scalar curvature is preserved under the $C^0$-convergence of metrics provided that the convergent sequence has a uniform Hölder bound. Moreover, if we consider the condition that the scalar curvature is bounded by some fixed continuous function from below in addition to the upper bound of the total scalar curvature, then such a condition is $C^0$-closed in the intersection of an arbitrarily fixed positive Yamabe conformal class and the space of metrics with a uniform Hölder bound.

18 pages