Paper · Oct 13, 2026

Curvature measures and volume preserving flows in h-convex geometry

Xinxiu Zhang

math.DGWider flowsarXiv:2610.18423
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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 -Gauss curvature flow for strictly h-convex hypersurfaces in hyperbolic space and prove that for any 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.

Comments: 37 pages. The paper was submitted on May 14, 2026

Other curvature flowsGauss curvature flowVolume-preserving & constrained flowsLong-time existence & convergence

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Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper Stability of the Volume Preserving Mean Curvature Flow in Hyperbolic Space (Zheng Huang, Longzhi Lin and Zhou Zhang, 2021)4Huang 2021 Volume preserving Gauss curvature flow of convex hypersurfaces in the hyperbolic space (Yong Wei, Bo Yang and Tailong Zhou, 2022)1Wei 2022 The chord Gauss curvature flow and its Lₚ chord Minkowski problem (Jinrong Hu, Yong Huang, Jian Lu et al., 2023)8Hu 2023 Volume preserving nonhomogeneous Gauss curvature flow in hyperbolic space (Yong Wei, Bo Yang and Tailong Zhou, 2024)2Wei 2024 Regularity of the p-Gauss curvature flow with flat side (Genggeng Huang, Xu-Jia Wang and Yang Zhou, 2024)7Huang 2024 Asymptotic behavior of a diffused interface volume-preserving mean curvature flow (Matteo Bonforte, Francesco Maggi and Daniel Restrepo, 2024)6Bonforte 2024 The Lₚ Gauss dual Minkowski problem (Na Fu and Jianping Sun, 2024)5Fu 2024 Modified support measures and Alexandrov-Fenchel type inequalities for nonsmooth h-convex bodies in hyperbolic space (Xinxiu Zhang, 2026)3Zhang 2026
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    Topics Gauss curvature flow, Volume-preserving & constrained flows, Long-time existence & convergence · Shared terms “curvature measure”, “hyperbolic space”, “geodesic sphere”

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    Topics Volume-preserving & constrained flows, Long-time existence & convergence · Shared terms “hyperbolic space”

  5. The Gauss dual Minkowski problem

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    Topics Gauss curvature flow, Other curvature flows · Shared terms “curvature measure”

  6. Asymptotic behavior of a diffused interface volume-preserving mean curvature flow

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  7. Regularity of the -Gauss curvature flow with flat side

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  8. The chord Gauss curvature flow and its chord Minkowski problem

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Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Other curvature flows Wider flows Flows that move a hypersurface or metric by some other function of curvature: Gauss curvature (worn stones), inverse mean curvature (the Penrose… Start with: Shapes of worn stones (Firey, 1974) Gauss curvature flow Wider flows Convex hypersurfaces shrink with normal speed a power K^α of their Gauss curvature. Firey proposed it as a model of stones worn by waves; for α>… Start with: Shapes of worn stones (Firey, 1974) Volume-preserving & constrained flows Wider flows Curvature flows with a constraint: subtract the average of the mean curvature and the enclosed volume stays fixed while area decreases, so spheres… Start with: On an area-preserving evolution equation for plane curves (Gage, 1986) Long-time existence & convergence A geometric flow either runs forever or stops because something blows up. Extension criteria say which quantity must blow up at a finite singular… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982)