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Curvature measures and volume preserving flows in h-convex geometry

  • Xinxiu Zhang
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Abstract

In this paper, we introduce a concept of modified curvature measures for h-convex domains in hyperbolic space and prove a shifted version of the generalized Alexandrov theorem, which characterizes geodesic balls in hyperbolic space via modified curvature measures. This result generalizes the classical work of Kohlmann (1998). We further investigate a volume-preserving shifted α-Gauss curvature flow for strictly h-convex hypersurfaces in hyperbolic space and prove that for any α>0 the solution exists for all time and converges smoothly to a geodesic sphere. The theory of modified curvature measures developed in the first part serves as a key ingredient in characterizing the Hausdorff limit shape of the flow.

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Cite

BibTeXRIS
BibTeX
@misc{zhang2026curvature,
  title = {{Curvature measures and volume preserving flows in h-convex geometry}},
  author = {Xinxiu Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.18423},
  eprint = {2610.18423},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.18423},
}
RIS
TY  - UNPB
TI  - Curvature measures and volume preserving flows in h-convex geometry
AU  - Xinxiu Zhang
PY  - 2026
DA  - 2026/10/13
PB  - arXiv
DO  - 10.48550/arxiv.2610.18423
UR  - https://arxiv.org/abs/2610.18423
AB  - In this paper, we introduce a concept of modified curvature measures for h-convex domains in hyperbolic space and prove a shifted version of the generalized Alexandrov theorem, which characterizes geodesic balls in hyperbolic space via modified curvature measures. This result generalizes the classical work of Kohlmann (1998). We further investigate a volume-preserving shifted α-Gauss curvature flow for strictly h-convex hypersurfaces in hyperbolic space and prove that for any α>0 the solution exists for all time and converges smoothly to a geodesic sphere. The theory of modified curvature measures developed in the first part serves as a key ingredient in characterizing the Hausdorff limit shape of the flow.
ER  -