Dynamical Stability of Translating Solitons to Mean Curvature Flow in Hyperbolic Space

We develop the theory of translating solitons for the Mean Curvature Flow (MCF) in hyperbolic space of dimension $n+1\ge 3$. More specifically, we establish that horospheres are dynamically stable as radial graphical solutions to MCF. To that end, we construct rotationally invariant translators analogous to the winglike solitons introduced by Clutterbuck, Schnürer and Schulze, which serve as barriers in an argument based on White’s avoidance principle and the strong maximum principle for parabolic PDEs.

💡 Research Summary

The paper studies translating solitons for the mean curvature flow (MCF) in hyperbolic space $\mathbb H^{n+1}$ with $n\ge 2$, focusing on the dynamical stability of horospheres when they are realized as radial graphical solutions. After recalling the basic setup of MCF in the upper‑half‑space model of hyperbolic space, the authors define a translating soliton as a hypersurface $\Sigma$ satisfying $H(p)=\langle p,N(p)\rangle$, where $H$ is the hyperbolic mean curvature, $N$ the unit normal, and $p$ the position vector. The simplest example is a horosphere $x_{n+1}=h$, which indeed fulfills the translator equation.

The authors then consider graphical solutions of the form $F(x,t)=e^{u(x,t)}x$, where $x\in S^n$ and $u$ satisfies a quasilinear parabolic PDE $\partial_t u = Q(u)$. By invoking a long‑time existence theorem of Unterberger (2023), they guarantee that for any locally Lipschitz initial data $u_0$ there exists a global graphical MCF solution.

A central technical contribution is the construction of a one‑parameter family of rotationally invariant translating solitons—called “translating catenoids”—that play the role of barriers. By writing a vertical rotational graph $X(\theta,s)=(s\phi(\theta),\varphi(s))$ and a horizontal rotational graph $x_1^2+\dots+x_n^2=d^2(x_{n+1})$, they derive two ordinary differential equations (10) and (13) whose solutions generate the vertical and horizontal pieces of a catenoid. Gluing these pieces yields embedded annular translators $\Sigma_r$ that lie between two horospheres $H_{r}^{-}$ and $H_{r}^{+}$. Theorem 3.3 establishes existence, asymptotic behavior as $r\to0$ (double copy of a horosphere) and as $r\to\infty$ (escape to infinity), and shows that no closed translator can exist in $\mathbb H^{n+1}$.

The main stability result (Theorem 4.1) states that if the initial graph $u_0$ differs from the horosphere graph $v_0$ by a function that decays to zero at infinity, then the corresponding MCF solution $u(x,t)$ converges uniformly to the translating horosphere $v(x,t)$ as $t\to\infty$. The proof proceeds in two steps. First, using the decay assumption, the authors find a large radius $R$ such that outside the ball $B_R$ the initial surfaces lie inside a thin slab $\Lambda_\varepsilon$. The translating catenoids $\Sigma_r^{\pm}$ are placed above and below this slab, and White’s avoidance principle guarantees that the evolving graphs remain trapped between them for all time, giving spatial convergence $|u-v|<\varepsilon$ outside $B_R$.

Second, they consider the difference $\omega=u-v$ and apply the strong maximum principle for parabolic equations. If $\omega$ attained a positive maximum at some interior point, the evolution inequality would contradict the barrier confinement, forcing $\omega$ to decay to zero uniformly in space as $t\to\infty$. Consequently, the horosphere is dynamically stable: any small $C^0$ perturbation of its graph disappears under the flow.

The paper concludes by mentioning analogous non‑radial translators (grim reapers) constructed in dimension three and poses open questions about their stability, suggesting that the barrier method may be adapted to other curvature flows in hyperbolic geometry. Overall, the work extends the Euclidean theory of translating solitons to the negatively curved setting and provides a robust analytical framework for studying stability of geometric flows in hyperbolic space.