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Aleksandrov reflection for Geometric Flows in Hyperbolic Spaces

Theodora Bourni, José M. Espinar, Aakash Mishra

TL;DR

This work develops a hyperbolic analogue of the Aleksandrov reflection technique for expanding curvature flows in $\mathbb{H}^{n+1}$ by marrying level-set (viscosity) formulations with reflection across totally geodesic hyperplanes. It yields robust graphical and Lipschitz estimates for level-set solutions, proves that compact-flow solutions become star-shaped after a finite waiting time and, when smooth, converge exponentially to an umbilic hypersurface at infinity; it further extends these results to non-compact hypersurfaces with a single ideal point by introducing horospherical reflection and barrier arguments, showing global graphical behavior and gradient control. Specializing to inverse-curvature flows, the paper provides a comprehensive compact-case theory (star-shapedness and exponential convergence to a geodesic sphere) and a non-compact horospherical theory (global graphs over horospheres and uniform gradient bounds), with long-time IMCF behavior culminating in convergence to a horosphere. Overall, the results deliver a flexible, quantitative framework for analyzing expanding flows in hyperbolic spaces from the dual weak- and strong-solution perspectives, enabling sharp asymptotics and broad initial-data applicability.

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. 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.

Aleksandrov reflection for Geometric Flows in Hyperbolic Spaces

TL;DR

This work develops a hyperbolic analogue of the Aleksandrov reflection technique for expanding curvature flows in by marrying level-set (viscosity) formulations with reflection across totally geodesic hyperplanes. It yields robust graphical and Lipschitz estimates for level-set solutions, proves that compact-flow solutions become star-shaped after a finite waiting time and, when smooth, converge exponentially to an umbilic hypersurface at infinity; it further extends these results to non-compact hypersurfaces with a single ideal point by introducing horospherical reflection and barrier arguments, showing global graphical behavior and gradient control. Specializing to inverse-curvature flows, the paper provides a comprehensive compact-case theory (star-shapedness and exponential convergence to a geodesic sphere) and a non-compact horospherical theory (global graphs over horospheres and uniform gradient bounds), with long-time IMCF behavior culminating in convergence to a horosphere. Overall, the results deliver a flexible, quantitative framework for analyzing expanding flows in hyperbolic spaces from the dual weak- and strong-solution perspectives, enabling sharp asymptotics and broad initial-data applicability.

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. 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.
Paper Structure (13 sections, 12 theorems, 101 equations, 7 figures)

This paper contains 13 sections, 12 theorems, 101 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hyperbolic space
  4. Aleksandrov reflection in hyperbolic space
  5. Level set flow
  6. Admissible foliations (compact case)
  7. Optimal admissible values (compact case).
  8. Optimal admissible value under the flow
  9. Applications: Inverse Curvature Flows
  10. Hypersurfaces with one point on the asymptotic boundary
  11. Admissible foliations (non-compact case)
  12. Optimal admissible values (non-compact case).
  13. Inverse curvature flows

Key Result

Lemma 3.5

Let $\Sigma$ be the zero set of a continuous function $u$. If $s$ is admissible for $u$ with respect to $\nu$, then $s$ is admissible for the triple $(\Sigma,\Omega,E)$ with respect to $\nu$, where $\Omega=\{u<0\}$ and $E=\{u>0\}$. If $u$ is the signed distance function, then the converse is also tr

Figures (7)

  • Figure 1: Admissibility of the value $s_0=0$
  • Figure 2: Monotonicity of $f_y$ along the geodesic $\gamma_y$.
  • Figure 3: Comparison on the equidistant hypersurface $\mathcal{E}_{\nu_0}(d_1)$.
  • Figure 4: Graphical representation of $\partial\Omega_t$ over a geodesic sphere with controlled gradient.
  • Figure 5: Gradient estimate for $\Sigma_t$ obtained via a totally geodesic comparison hypersurface
  • ...and 2 more figures

Theorems & Definitions (30)

  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Remark 3.4
  • Lemma 3.5
  • proof
  • Theorem 3.6
  • proof
  • Proposition 3.7
  • proof
  • ...and 20 more