Aleksandrov reflection for Geometric Flows in Hyperbolic Spaces

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. As a consequence, we obtain graphical and Lipschitz estimates. Using these estimates, we show that solutions become starshaped and therefore converge exponentially fast to an umbilic hypersurface at infinity. We also extend our results to the non-compact setting, assuming that the solution has a unique point at infinity. In this case, we prove that the flow becomes a graph over a horosphere with uniform gradient bounds and converges to a limiting horosphere.

💡 Research Summary

The paper develops an Aleksandrov reflection framework for a broad class of expanding extrinsic curvature flows in hyperbolic space ( \mathbb H^{n+1} ), with the inverse mean curvature flow (IMCF) serving as the primary model. The authors work in the weak, viscosity‑type level‑set setting, which allows them to treat initial hypersurfaces that are merely continuous, possibly non‑convex, and even non‑compact.

First, the authors recall the geometry of hyperbolic space in two concrete models—the Poincaré ball and the upper half‑space—describing the four families of isoparametric hypersurfaces (geodesic spheres, totally geodesic hyperplanes, equidistant hypersurfaces, and horospheres). These serve as canonical barriers throughout the analysis.

The Aleksandrov reflection method is then adapted to hyperbolic space. Given a totally geodesic hyperplane (P), the isometric involution (R_P) reflects points across (P). By moving a family of such hyperplanes monotonically toward a hypersurface (\Sigma), one records the first contact point (s_0) and then reflects the portion of (\Sigma) lying on one side of (P). A second contact point (s_1) is identified, and the monotonicity of the flow’s speed function (F) (increasing in each principal curvature) together with the maximum principle yields symmetry or comparison information.

In the level‑set formulation, the evolving hypersurface is the zero‑level set of a continuous function (u(x,t)) solving the degenerate parabolic PDE
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