Curvature integral, volume ratio, bi-Lipschitz for spaces with curvature bounded below
Abstract
By the paper \cite{Pet2009upper}, \cite{LiNan2026}, we know that there exists C(n)>0, for any complete compact or non-compact Riemannian n manifold M with non-negative sectional curvature, without boundary and any R>0, R2−n∫B(p,R) Scal ≤ C(n). Based on this result, we will prove that (1) There exists constant C(n). If M is a complete, n dimensional, non-compact Riemannian manifold with non-negative sectional curvature, then for any p∈ M, R>0, R2−n∫B(p,R) Scal dvol ≤ C(n)(1−v(M)), where Scal is the scalar curvature and v(M)=limR→ ∞ volB(p,R)/volB(0,R) is the asymptotic volume ratio. (2) There exists constant C(n). If M is a complete, n dimensional Riemannian manifold with sectional curvature ≥ 1, then ∫M (Scal−n(n−1)) dvol ≤ C(n)(1−vol(M)/vol(Sn(1))).
Related
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BibTeX
@misc{jiang2026curvature,
title = {{Curvature integral, volume ratio, bi-Lipschitz for spaces with curvature bounded below}},
author = {Yin Jiang and Nan Li},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.29982},
eprint = {2609.29982},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.29982},
}
RIS
TY - UNPB
TI - Curvature integral, volume ratio, bi-Lipschitz for spaces with curvature bounded below
AU - Yin Jiang
AU - Nan Li
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.29982
UR - https://arxiv.org/abs/2609.29982
AB - By the paper \cite{Pet2009upper}, \cite{LiNan2026}, we know that there exists C(n)>0, for any complete compact or non-compact Riemannian n manifold M with non-negative sectional curvature, without boundary and any R>0, R^2−n∫_B(p,R) Scal ≤ C(n). Based on this result, we will prove that (1) There exists constant C(n). If M is a complete, n dimensional, non-compact Riemannian manifold with non-negative sectional curvature, then for any p∈ M, R>0, R^2−n∫_B(p,R) Scal dvol ≤ C(n)(1−v(M)), where Scal is the scalar curvature and v(M)=lim_R→ ∞ volB(p,R)/volB(0,R) is the asymptotic volume ratio. (2) There exists constant C(n). If M is a complete, n dimensional Riemannian manifold with sectional curvature ≥ 1, then ∫_M (Scal−n(n−1)) dvol ≤ C(n)(1−vol(M)/vol(S^n(1))).
ER -