TY - JOUR
T1 - An Alternative Proof for the Upper Bound of Curvature Integral on Manifolds with Lower Sectional Curvature Bound
AU - Li , Nan
JO - Journal of Mathematical Study
VL - 2
SP - 164
EP - 174
PY - 2025
DA - 2025/06
SN - 58
DO - http://doi.org/10.4208/jms.v58n2.25.03
UR - https://global-sci.org/intro/article_detail/jms/24208.html
KW - Scalar curvature, sectional curvature, $L^1$-norm, Alexandrov space, stratification, covering.
AB - <p style="text-align: justify;">Petrunin proved that the integral of scalar curvature in a unit ball is bounded
from above in terms of the dimension of the manifold and the lower bound of the sectional curvature. In this paper, we give an alternative proof for this result. The main
difference between this proof and Petrunin’s original proof is that we construct a stratified finite covering and apply it directly to the given manifold, rather than arguing by
contradiction for a sequence of manifolds, which satisfy some technical lifting properties.</p>