Sharp integral bound of scalar curvature on $3$-manifolds

TL;DR

The paper resolves the 3D case of bounding the average scalar curvature on large geodesic balls in complete manifolds with nonnegative Ricci curvature by establishing an asymptotic bound of $8\pi$. The authors construct a smooth distance-like function $\rho$ satisfying $\Delta \rho = -1 + |\nabla \rho|^2$ and analyze its level sets, decomposing them into an unbounded boundary $\ell_0(s)$ and finite fingers $\ell_1(s)$; a Gauss-Bonnet argument together with a key Ricci-related identity connects curvature information to ball-integrals. A key technical ingredient is controlling the size of fingers via a $\,\mu$-bubble diameter estimate and excess-function methods, which together with Bochner-type identities yield a bound on $\int_{B_p(R)} S$ of the form $8\pi R$ plus lower-order terms. This method relies only on Ricci nonnegativity and bounds on $S$, and provides a robust approach that could extend to more general curvature-function bounds, informing the distribution of scalar curvature on noncompact 3-manifolds. Overall, the work sharpens our understanding of how nonnegative Ricci curvature constrains the global curvature distribution in dimension three and yields a sharp, scale-invariant bound with potential for further generalization.

Abstract

It is shown that the integral of the scalar curvature on a geodesic ball of radius $R$ in a three-dimensional complete manifold with nonnegative Ricci curvature is bounded above by $8πR$ asymptotically for large $R$ provided that the scalar curvature is bounded between two positive constants.

Theorems & Definitions (21)

• Theorem 1.2
• Conjecture 1.3
• Theorem 1.4
• Lemma 1.5
• Remark 1.6
• Proposition 2.1
• proof
• Lemma 3.1
• Proposition 3.2
• Lemma 3.3