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Stability of the Catenoid for the Hyperbolic Vanishing Mean Curvature Equation Outside Symmetry

Jonas Luhrmann, Sung-Jin Oh, Sohrab Shahshahani

TL;DR

This work proves nonlinear asymptotic stability of the Lorentzian catenoid as a stationary solution to the HVMC equation in Minkowski space for dimensions n≥5, under a codimension-1 initial-data constraint and modulo translation and Lorentz boosts. The authors introduce a moving catenoid profile and a robust modulation framework to track translation and boost parameters, accompanied by a moving hyperboloidal foliation and a novel profile construction. Central to the analysis are integrated local energy decay (ILED) and a refined r^p-vectorfield strategy adapted to the moving profile, along with a first-order formulation and a smoothing mechanism to handle quasilinear effects and spectral obstructions. The result extends radial stability results to a non-symmetric, higher-dimensional setting and offers techniques potentially applicable to other quasilinear soliton stability problems, with implications for stability analyses of topological solitons in nonlinear wave equations.

Abstract

We study the problem of stability of the catenoid, which is an asymptotically flat rotationally symmetric minimal surface in Euclidean space, viewed as a stationary solution to the hyperbolic vanishing mean curvature equation in Minkowski space. The latter is a quasilinear wave equation that constitutes the hyperbolic counterpart of the minimal surface equation in Euclidean space. Our main result is the nonlinear asymptotic stability, modulo suitable translation and boost (i.e., modulation), of the $n$-dimensional catenoid with respect to a codimension one set of initial data perturbations without any symmetry assumptions, for $n \geq 5$. The modulation and the codimension one restriction on the data are necessary and optimal in view of the kernel and the unique simple eigenvalue, respectively, of the stability operator of the catenoid. In a broader context, this paper fits in the long tradition of studies of soliton stability problems. From this viewpoint, our aim here is to tackle some new issues that arise due to the quasilinear nature of the underlying hyperbolic equation. Ideas introduced in this paper include a new profile construction and modulation analysis to track the evolution of the translation and boost parameters of the stationary solution, a new scheme for proving integrated local energy decay for the perturbation in the quasilinear and modulation-theoretic context, and an adaptation of the vectorfield method in the presence of dynamic translations and boosts of the stationary solution.

Stability of the Catenoid for the Hyperbolic Vanishing Mean Curvature Equation Outside Symmetry

TL;DR

This work proves nonlinear asymptotic stability of the Lorentzian catenoid as a stationary solution to the HVMC equation in Minkowski space for dimensions n≥5, under a codimension-1 initial-data constraint and modulo translation and Lorentz boosts. The authors introduce a moving catenoid profile and a robust modulation framework to track translation and boost parameters, accompanied by a moving hyperboloidal foliation and a novel profile construction. Central to the analysis are integrated local energy decay (ILED) and a refined r^p-vectorfield strategy adapted to the moving profile, along with a first-order formulation and a smoothing mechanism to handle quasilinear effects and spectral obstructions. The result extends radial stability results to a non-symmetric, higher-dimensional setting and offers techniques potentially applicable to other quasilinear soliton stability problems, with implications for stability analyses of topological solitons in nonlinear wave equations.

Abstract

We study the problem of stability of the catenoid, which is an asymptotically flat rotationally symmetric minimal surface in Euclidean space, viewed as a stationary solution to the hyperbolic vanishing mean curvature equation in Minkowski space. The latter is a quasilinear wave equation that constitutes the hyperbolic counterpart of the minimal surface equation in Euclidean space. Our main result is the nonlinear asymptotic stability, modulo suitable translation and boost (i.e., modulation), of the -dimensional catenoid with respect to a codimension one set of initial data perturbations without any symmetry assumptions, for . The modulation and the codimension one restriction on the data are necessary and optimal in view of the kernel and the unique simple eigenvalue, respectively, of the stability operator of the catenoid. In a broader context, this paper fits in the long tradition of studies of soliton stability problems. From this viewpoint, our aim here is to tackle some new issues that arise due to the quasilinear nature of the underlying hyperbolic equation. Ideas introduced in this paper include a new profile construction and modulation analysis to track the evolution of the translation and boost parameters of the stationary solution, a new scheme for proving integrated local energy decay for the perturbation in the quasilinear and modulation-theoretic context, and an adaptation of the vectorfield method in the presence of dynamic translations and boosts of the stationary solution.
Paper Structure (57 sections, 31 theorems, 614 equations)

This paper contains 57 sections, 31 theorems, 614 equations.

Table of Contents

  1. Introduction
  2. Stability Problems for the Hyperbolic Vanishing Mean Curvature Equation
  3. The Catenoid Solution
  4. First Statement of the Main Result
  5. Overall Scheme and Main Difficulties
  6. Overall scheme
  7. Main difficulties
  8. Profile, Foliation and Gauge Construction
  9. First-Order Formulation, Modulation Equations and Selection of a Codimension One Set of Initial Data
  10. Uniform Boundedness of Energy and Integrated Local Energy Decay
  11. Vectorfield Method for Moving Profile
  12. Second Statement of the Main Result
  13. Further Discussions
  14. Other prior works on the hyperbolic vanishing mean curvature equation
  15. Comparison with black hole stability
  16. ...and 42 more sections

Key Result

Theorem 1.2

Let $\Phi_0\colon I\times \mathbb S^{n-1} \to\{0\}\times\mathbb R^{n+1}$, $n\geq 5$, be an embedding and $\Phi_1\colon I\times \mathbb S^{n-1}\to \mathbb R^{1+(n+1)}$ be a family of future directed timelike vectors such that $\Phi_0=F$ and $\Phi_1=(1,0)$ outside of a compact set. Suppose $\Phi_0$ an

Theorems & Definitions (76)

  • Definition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.5
  • Remark 1.6
  • Remark 1.7
  • Proposition 2.1
  • Lemma 2.2
  • proof
  • ...and 66 more