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

TL;DR

This work analyzes translating solitons for mean curvature flow in hyperbolic space $\mathbb{H}^{n+1}$ and proves horospheres are dynamically stable as graphical translating solutions. The authors develop a barrier framework by constructing translating catenoids and applying White's avoidance principle together with the strong maximum principle to obtain convergence results, extending Euclidean stability insights to the hyperbolic setting. They also introduce grim reaper-type translators in $\mathbb{H}^{n+1}$, establishing a hyperbolic analogue of soliton barriers and highlighting open questions about their stability. Overall, the paper provides new tools for understanding long-time behavior of graphical MCF in negatively curved spaces and sets the stage for further stability analyses of hyperbolic translators.

Abstract

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.

Figures (3)

• Figure 1: The profile curve of a translating catenoid $\varSigma_r$ in $\mathbb{H} ^{n+1}$ between (and asymptotic to) two horospheres $\mathscr H_{r^-}$ and $\mathscr H_{r^+}$. $\varSigma_r\setminus \mathscr H$ decomposes as two vertical graphs $\varSigma_r^-$ and $\varSigma_r^+$ over the complement of the Euclidean ball of radius $r$ centered at the rotation axis $x_{n+1}$ in the horosphere $\mathscr H$.
• Figure 2: Translating catenoids acting as barriers: $\varSigma^+\subset \varLambda_\epsilon^+$ and $\varSigma^-\subset \varLambda_\epsilon^-$ are a positive distance from the graph $F(\mathcal{S}\times\{0\})$, whose part outside $\mathscr C_{R_1}$ lies in the slab $\varLambda_\epsilon$. As the surfaces flow under the MCF, $F_t(\mathcal{S}\setminus \mathscr C_R)$ remains between the translating catenoids $\Gamma_t(\varSigma^-)$ and $\Gamma_t(\varSigma^+)$.
• Figure 3: The profile curve of a solution $\phi$ to \ref{['eq-EDOparabolic']} with $\lambda = \phi'(0) >0$.

Theorems & Definitions (16)

• Example 2.1
• Definition 2.2
• Theorem 2.3
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Theorem 3.3
• Remark 3.4
• Theorem 4.1