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 ∫Bp(r)Sdμg ≤ 2n(n−1)ωn (1−v1/(n−1))rn−2, when p is the rotational origin. Here ωn is the volume of the Euclidean unit ball and v=VMn is the asymptotic volume ratio. In dimension three, with the center at a pole, we obtain a sharp bound ∫Bp(r)Sdμg≤16πr, without any positive volume growth assumption. We also obtain a bound for every asymptotic volume ratio which is sharp for 0≤ v≤1/4. Under nonnegative sectional curvature and v>0, we prove the explicit bound at arbitrary centers ∫Bp(r)Sdμg≤12π(1−v)v−3r. If 0≤1−v≤1/100, we further obtain ∫Bp(r)Sdμg≤[8π(1−v)+200π(1−v)2]r. The leading coefficient 8π is sharp, and a remainder of quadratic order in 1−v is necessary.
Related
- Rigidity of Constant-Scalar-Curvature Ricci Shrinkers with Totally Geodesic Focal Set
- Unbounded normalized scalar curvature integrals in dimension four
- Scalar--Mean Rigidity for Hyperbolic Balls in Dimensions 2≤ n≤ 5
- Ergodicity of the geodesic flow on surfaces of nonpositive curvature
- Integral curvature estimates and geodesic limits for stable abelian Higgs fields
BibTeX
@misc{hu2026scalar,
title = {{Scalar curvature integrals on geodesic balls}},
author = {Qixuan Hu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.29596},
eprint = {2610.29596},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.29596},
}
RIS
TY - UNPB
TI - Scalar curvature integrals on geodesic balls
AU - Qixuan Hu
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.29596
UR - https://arxiv.org/abs/2610.29596
AB - We study scalar curvature integrals over geodesic balls on complete noncompact manifolds. Under nonnegative Ricci curvature and rotational symmetry, we prove the sharp bound ∫_B_p(r)Sdμ_g ≤ 2n(n−1)ω_n (1−v^1/(n−1))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 ∫_B_p(r)Sdμ_g≤16πr, without any positive volume growth assumption. We also obtain a bound for every asymptotic volume ratio which is sharp for 0≤ v≤1/4. Under nonnegative sectional curvature and v>0, we prove the explicit bound at arbitrary centers ∫_B_p(r)Sdμ_g≤12π(1−v)v^−3r. If 0≤1−v≤1/100, we further obtain ∫_B_p(r)Sdμ_g≤[8π(1−v)+200π(1−v)^2]r. The leading coefficient 8π is sharp, and a remainder of quadratic order in 1−v is necessary.
ER -