Paper · Feb 12, 2026
Aleksandrov reflection for Geometric Flows in Hyperbolic Spaces
As on arXiv
Abstract
We develop an Aleksandrov reflection framework for a large class of expanding curvature flows in hyperbolic space, with inverse mean curvature flow serving as a model case. The method applies to the level-set formulation of the flow, and as a consequence we obtain graphical and Lipschitz estimates. Using these estimates, we show that solutions become star-shaped and therefore converge exponentially fast to an umbilic hypersurface at infinity. We also extend these results to the non-compact setting in two cases. First, assuming the asymptotic boundary of the solution consists of a single point, we show that the flow becomes a graph over a horosphere with uniform gradient bounds and converges to a limiting horosphere. Second, assuming the asymptotic boundary consists of two points, we prove that the flow eventually becomes a global graph over a hyperbolic cylinder with uniform gradient bounds; this is achieved through an explicit cylindrical barrier construction analogous to the horospherical one.
Comments: The major update in this version is the treatment of non-compact surfaces with two points at infinity, extending the previous analysis, which was restricted to a single point at infinity. 10 figures
Its neighbourhood
Connected papers
The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2021, plus the canon.
-
On weak inverse mean curvature flow and Minkowski-type inequalities in hyperbolic space
Topics Inverse mean curvature flow, Weak & singular flows, Boundaries & non-compact · Shared terms “hyperbolic space”, “reflection”, “star shaped”
-
A Minkowski type inequality in warped cylinders
Topics Inverse mean curvature flow, Weak & singular flows, Boundaries & non-compact · Shared terms “hyperbolic space”, “star shaped”
-
The streamlines of -harmonic functions obey the inverse mean curvature flow
Topics Inverse mean curvature flow, Weak & singular flows, Other curvature flows · Shared terms “set formulation”, “level set”
-
Global weak solutions for the inverse mean curvature flow in the Heisenberg group
Topics Inverse mean curvature flow, Weak & singular flows, Other curvature flows · Shared terms “level set”, “Lipschitz”, “formulation”
-
Inverse mean curvature flow with outer obstacle
Topics Inverse mean curvature flow, Weak & singular flows, Boundaries & non-compact · Shared terms “level set”
-
Infinity-harmonic functions and inverse mean curvature flow clusters
Topics Inverse mean curvature flow, Weak & singular flows, Boundaries & non-compact
-
On the evolution of hypersurfaces along their inverse spacetime mean curvature
Topics Inverse mean curvature flow, Weak & singular flows, Boundaries & non-compact
-
Inverse mean curvature flow with a free boundary in hyperbolic space
Topics Inverse mean curvature flow, Other curvature flows · Shared terms “hyperbolic space”
Read around it
Background reading
The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.
In the guidebook
- 12C.9After Perelman
Singular flows, the Smale conjecture, and Bamler’s theory.
- 6A.10Curve Shortening and the First Geometric Flows
Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.