PreprintarXiv licencearXiv
Convex-like Submanifolds, Caps and a Volume Comparison Theorem
Abstract
Convex-like submanifolds in the Lorentz-Minkowski spaces are introduced in this paper as a natural and broad generalization of convex surfaces in the Euclidean space. Associated with such a convex-like submanifold in a Minkowski space, we define roofs (which resembles the supporting planes), caps (which are generated by the light rays in the normal bundle), and towers (which revives the notion of interior for such high codimensional objects). As a touchstone of this theory, a volume comparison theorem is proved, which is a non-trivial generalization of the obvious fact that among two nested convex surfaces, the outer one has larger volume. Finally we provide two important applications of this theorem.
Related
- A very short proof of the subgaussian convex-order comparison theorem
- Relative volume comparison theorem under Kato type conditions
- Arithmeticity, superrigidity, and totally geodesic submanifolds of infinite-volume hyperbolic manifolds
- An optimal pinching theorem on compact minimal submanifolds in the Euclidean spheres via eigenvalues of fundamental matrices
- A sharp pinching theorem for closed minimal Legendrian submanifolds
BibTeX
@misc{hu2026convexlike,
title = {{Convex-like Submanifolds, Caps and a Volume Comparison Theorem}},
author = {Baichuan Hu and Xiang Ma and Dong Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.31332},
eprint = {2610.31332},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.31332},
}
RIS
TY - UNPB
TI - Convex-like Submanifolds, Caps and a Volume Comparison Theorem
AU - Baichuan Hu
AU - Xiang Ma
AU - Dong Zhang
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.31332
UR - https://arxiv.org/abs/2610.31332
AB - Convex-like submanifolds in the Lorentz-Minkowski spaces are introduced in this paper as a natural and broad generalization of convex surfaces in the Euclidean space. Associated with such a convex-like submanifold in a Minkowski space, we define roofs (which resembles the supporting planes), caps (which are generated by the light rays in the normal bundle), and towers (which revives the notion of interior for such high codimensional objects). As a touchstone of this theory, a volume comparison theorem is proved, which is a non-trivial generalization of the obvious fact that among two nested convex surfaces, the outer one has larger volume. Finally we provide two important applications of this theorem.
ER -