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Unbounded normalized scalar curvature integrals in dimension four
Abstract
We give counterexamples to Yau's question on normalized scalar curvature integrals in every dimension n≥4. For each such n, there exists a smooth complete Riemannian metric g on ℝn with nonnegative Ricci curvature and a pole such that limR→∞R2−n∫Bq(R)Scalg dVg=+∞ for every fixed q∈ℝn. Here Bq(R) denotes the geodesic ball of radius R centered at q, and Scalg is the scalar curvature of g.
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BibTeX
@misc{cheng2026unbounded,
title = {{Unbounded normalized scalar curvature integrals in dimension four}},
author = {Haoxuan Cheng},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.10323},
eprint = {2609.10323},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.10323},
}
RIS
TY - UNPB
TI - Unbounded normalized scalar curvature integrals in dimension four
AU - Haoxuan Cheng
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.10323
UR - https://arxiv.org/abs/2609.10323
AB - We give counterexamples to Yau's question on normalized scalar curvature integrals in every dimension n≥4. For each such n, there exists a smooth complete Riemannian metric g on ℝ^n with nonnegative Ricci curvature and a pole such that lim_R→∞R^2−n∫_B_q(R)Scal_g dV_g=+∞ for every fixed q∈ℝ^n. Here B_q(R) denotes the geodesic ball of radius R centered at q, and Scal_g is the scalar curvature of g.
ER -