Scalar curvature integrals on geodesic balls

Abstract

We study scalar curvature integrals over geodesic balls on complete noncompact manifolds. Under nonnegative Ricci curvature and rotational symmetry, we prove the sharp bound \[ \int_{B_p(r)}S\text{d}μ_g \le 2n(n-1)ω_n \bigl(1-v^{1/(n-1)}\bigr)r^{n-2}, \] when $p$ is the rotational origin. Here $ω_n$ is the volume of the Euclidean unit ball and $v=V_{M^n}$ is the asymptotic volume ratio. In dimension three, with the center at a pole, we obtain a sharp bound \[ \int_{B_p(r)}S\text{d}μ_g\le16πr, \] without any positive volume growth assumption. We also obtain a bound for every asymptotic volume ratio which is sharp for $0\le v\le1/4$. Under nonnegative sectional curvature and $v>0$, we prove the explicit bound at arbitrary centers \[ \int_{B_p(r)}S\text{d}μ_g\le12π(1-v)v^{-3}r. \] If $0\le1-v\le1/100$, we further obtain \[ \int_{B_p(r)}S\text{d}μ_g\le\bigl[8π(1-v)+200π(1-v)^2\bigr]r. \] The leading coefficient $8π$ is sharp, and a remainder of quadratic order in $1-v$ is necessary.