Scalar curvature growth on nonnegatively curved three-manifolds
Abstract
Let (M3,g) be a complete, connected, noncompact Riemannian three-manifold without boundary and with nonnegative sectional curvature. We prove that its scalar-curvature integral over geodesic balls, divided by the radius, has a limit. In the one-ended case, limr→∞1/r∫Bp(r)Scal d V =8π(χ(M)−VM)≤8π(1−VM), where VM is the asymptotic volume ratio. In the one-ended case, we show that χ(M)∈{0,1}. Thus positive asymptotic volume ratio gives the value 8π(1−VM), while in the collapsed case the value is determined by the topology of the end. No pole or scalar-curvature bound is assumed. The proof combines separate smooth approximations of the Busemann and distance functions, integrable negative curvature errors, and a determinant estimate on the level surfaces. An averaged boundary estimate then gives convergence of the integrated extrinsic curvature. In the two-ended case, the splitting theorem gives the exact limit 8πχ(N) for the compact surface factor N. The one-ended upper bound is attained for every prescribed asymptotic volume ratio in [0,1], and the two-ended bound is also sharp.
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BibTeX
@misc{hu2026scalar,
title = {{Scalar curvature growth on nonnegatively curved three-manifolds}},
author = {Qixuan Hu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.26717},
eprint = {2609.26717},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.26717},
}
RIS
TY - UNPB
TI - Scalar curvature growth on nonnegatively curved three-manifolds
AU - Qixuan Hu
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.26717
UR - https://arxiv.org/abs/2609.26717
AB - Let (M^3,g) be a complete, connected, noncompact Riemannian three-manifold without boundary and with nonnegative sectional curvature. We prove that its scalar-curvature integral over geodesic balls, divided by the radius, has a limit. In the one-ended case, lim_r→∞1/r∫_B_p(r)Scal d V =8π(χ(M)−V_M)≤8π(1−V_M), where V_M is the asymptotic volume ratio. In the one-ended case, we show that χ(M)∈{0,1}. Thus positive asymptotic volume ratio gives the value 8π(1−V_M), while in the collapsed case the value is determined by the topology of the end. No pole or scalar-curvature bound is assumed. The proof combines separate smooth approximations of the Busemann and distance functions, integrable negative curvature errors, and a determinant estimate on the level surfaces. An averaged boundary estimate then gives convergence of the integrated extrinsic curvature. In the two-ended case, the splitting theorem gives the exact limit 8πχ(N) for the compact surface factor N. The one-ended upper bound is attained for every prescribed asymptotic volume ratio in [0,1], and the two-ended bound is also sharp.
ER -