Extension of the sharp Hessian inequality beyond Einstein manifolds

Investigate whether the sharp Hessian inequality for the Green's function power on closed manifolds with positive sectional curvature remains valid without assuming that the manifold is Einstein.

Background

The paper proves that if a closed, connected Riemannian manifold is Einstein and has sectional curvature bounded below by a positive constant, then the Green's function power s = G{1/(2-n)} associated with the operator -Δ + n(n−2)k/4 satisfies the sharp quadratic-form inequality Hess(s²) + ks²g ≤ 2g. The proof relies analytically on parallel Ricci curvature, which is equivalent to the Einstein condition in the setting considered.

The authors explicitly indicate that they do not expect this sharp estimate to remain valid without the Einstein assumption. Consequently, determining whether an analogue can hold for positively curved, non-Einstein manifolds—or identifying a precise counterexample or necessary replacement hypothesis—remains unresolved in the stated discussion.

References

In this sense, we do not expect Theorem \ref{intro-th: sharp hessian estimate} to hold without this additional assumption.

— Sharp Hessian inequalities and monotonicity for Green's functions of positively curved Einstein manifolds  (2610.11295 - Manea et al., 8 Oct 2026) in Section 1, Introduction, subsection “The sharp Hessian inequality”