L2 Curvature Bounds on Ricci flows with bounded scalar curvature up to dimension 7
Abstract
In this paper, we study Ricci flows with uniformly bounded scalar curvature. On such Ricci flows we establish uniform L2 Riemannian curvature bounds and L4 Ricci curvature bounds up to dimension 7. These estimates improves L2−ε curvature bounds established by Bamler and L2 curvature bounds in dimension 4 established by Bamler and Zhang. More generally, we establish uniform L2 curvature bounds on n-manifolds satisfying an integral Lp Ricci curvature bound for p> n/2, a uniform volume lower bound, and a strong ε-regularity assumption. This result extends the celebrated L2 curvature estimates of Jiang and Naber, which were established under pointwise two-sided Ricci curvature bounds. To overcome the lack of pointwise control, we develop an Lp adaptation of the neck decomposition theorem and establish a superconvexity estimate for the curvature on neck regions. Furthermore, we demonstrate that our structural assumptions are dynamically natural on Ricci flows with bounded scalar curvature. As an application, by combining our estimate with the a priori L4−ε curvature radius bound of Ricci flow established by Bamler, we obtain uniform L2 curvature bounds on Ricci flows with bounded scalar curvature of dimension n ≤ 7.
Related
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- Eigenvalue comparison theorems for the Witten-Laplacian and the weighted p-Laplacian on complete manifolds with a modified Ricci curvature bounded from below
BibTeX
@misc{huang2026curvature,
title = {{$L^2$ Curvature Bounds on Ricci flows with bounded scalar curvature up to dimension 7}},
author = {Xianbin Huang and Wancheng Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.10575},
eprint = {2610.10575},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.10575},
}
RIS
TY - UNPB
TI - L^2 Curvature Bounds on Ricci flows with bounded scalar curvature up to dimension 7
AU - Xianbin Huang
AU - Wancheng Zhang
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.10575
UR - https://arxiv.org/abs/2610.10575
AB - In this paper, we study Ricci flows with uniformly bounded scalar curvature. On such Ricci flows we establish uniform L^2 Riemannian curvature bounds and L^4 Ricci curvature bounds up to dimension 7. These estimates improves L^2−ε curvature bounds established by Bamler and L^2 curvature bounds in dimension 4 established by Bamler and Zhang. More generally, we establish uniform L^2 curvature bounds on n-manifolds satisfying an integral L^p Ricci curvature bound for p> n/2, a uniform volume lower bound, and a strong ε-regularity assumption. This result extends the celebrated L^2 curvature estimates of Jiang and Naber, which were established under pointwise two-sided Ricci curvature bounds. To overcome the lack of pointwise control, we develop an L^p adaptation of the neck decomposition theorem and establish a superconvexity estimate for the curvature on neck regions. Furthermore, we demonstrate that our structural assumptions are dynamically natural on Ricci flows with bounded scalar curvature. As an application, by combining our estimate with the a priori L^4−ε curvature radius bound of Ricci flow established by Bamler, we obtain uniform L^2 curvature bounds on Ricci flows with bounded scalar curvature of dimension n ≤ 7.
ER -