Sharp total scalar-curvature bound under sectional-curvature lower bounds

Establish whether the unit round sphere gives the sharp upper bound for the total scalar curvature of manifolds satisfying the sectional-curvature condition \(\sec_g\ge1\).

Background

The introduction reviews comparison results that bound total scalar curvature under lower sectional-curvature assumptions and notes that the optimal bound was not known in the general sectional-curvature setting. It identifies a conjecture attributed to Li asserting that the unit round sphere realizes the sharp bound when $\sec_g\ge1$. The present paper proves the analogous sharp estimate under the stronger assumption that the curvature operator satisfies $R\ge\operatorname{Id}$, so the broader sectional-curvature problem remains distinct from the result established here.