Curvature integral on convex hypersurfaces
Abstract
Let Mn, n≥3, be the smooth, connected, complete, noncompact boundary of a closed convex set with nonempty interior in ℝn+1. We prove that its normalized scalar-curvature integral at infinity is at most 4πωn−2(1−AVR(M)), except when, up to an ambient rigid motion, M is the product of a smooth closed convex surface in ℝ3 and ℝn−2. These exceptional products have zero asymptotic volume ratio and limit 8πωn−2. We also characterize the equality case for the 4πωn−2(1−AVR(M)) bound. The proof uses exterior tube volumes together with the cone Steiner and Gauss--Bonnet formulas. An elementary comparison of intrinsic and Euclidean radial distances proves existence of the intrinsic limit and identifies its exact value from the cone coefficients. The same conclusions extend to orientable convex submanifolds of arbitrary codimension
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BibTeX
@misc{hu2026curvature,
title = {{Curvature integral on convex hypersurfaces}},
author = {Qixuan Hu and Shuai Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.03490},
eprint = {2610.03490},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.03490},
}
RIS
TY - UNPB
TI - Curvature integral on convex hypersurfaces
AU - Qixuan Hu
AU - Shuai Zhang
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.03490
UR - https://arxiv.org/abs/2610.03490
AB - Let M^n, n≥3, be the smooth, connected, complete, noncompact boundary of a closed convex set with nonempty interior in ℝ^n+1. We prove that its normalized scalar-curvature integral at infinity is at most 4πω_n−2(1−AVR(M)), except when, up to an ambient rigid motion, M is the product of a smooth closed convex surface in ℝ^3 and ℝ^n−2. These exceptional products have zero asymptotic volume ratio and limit 8πω_n−2. We also characterize the equality case for the 4πω_n−2(1−AVR(M)) bound. The proof uses exterior tube volumes together with the cone Steiner and Gauss--Bonnet formulas. An elementary comparison of intrinsic and Euclidean radial distances proves existence of the intrinsic limit and identifies its exact value from the cone coefficients. The same conclusions extend to orientable convex submanifolds of arbitrary codimension
ER -