PreprintCC BY 4.0arXiv
Rigidity and volume pinching for the sharp gradient estimate in positive Ricci curvature
Abstract
Colding established a sharp gradient estimate for the Green function on manifolds with non-negative Ricci curvature, which was extended to positive Ricci curvature by Manea recently. We prove almost rigidity of such gradient estimates for positive Ricci curvature. We show that the average of the gradient deficit, namely fint (1−|∇ b|2) dV, controls the volume deficit in an optimal quantitative manner; here b is the distance-like function defined using the Green function. This implies a quantitative almost rigidity theorem for manifolds with ric≥(n−1)g and a gap theorem for Einstein manifolds. We also prove a rigidity theorem for closed 4-dimensional Einstein manifolds by finding a new monotonicity formula.
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BibTeX
@misc{kim2026rigidity,
title = {{Rigidity and volume pinching for the sharp gradient estimate in positive Ricci curvature}},
author = {Junyoung Kim and Jiewon Park},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.03739},
eprint = {2609.03739},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.03739},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Rigidity and volume pinching for the sharp gradient estimate in positive Ricci curvature
AU - Junyoung Kim
AU - Jiewon Park
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2609.03739
UR - https://arxiv.org/abs/2609.03739
AB - Colding established a sharp gradient estimate for the Green function on manifolds with non-negative Ricci curvature, which was extended to positive Ricci curvature by Manea recently. We prove almost rigidity of such gradient estimates for positive Ricci curvature. We show that the average of the gradient deficit, namely fint (1−|∇ b|^2) dV, controls the volume deficit in an optimal quantitative manner; here b is the distance-like function defined using the Green function. This implies a quantitative almost rigidity theorem for manifolds with ric≥(n−1)g and a gap theorem for Einstein manifolds. We also prove a rigidity theorem for closed 4-dimensional Einstein manifolds by finding a new monotonicity formula.
ER -