[Paper Review] Asymptotic stability of harmonic maps between 2D hyperbolic spaces under the wave map equation. II. Small energy case

Ze Li, Xiao Ma+1 more|arXiv (Cornell University)|Mar 15, 2017

Advanced Mathematical Physics Problems75 references3 citations

TL;DR

This paper establishes the asymptotic stability of small energy harmonic maps from $$\mathbb{H}^2$$ to $$\mathbb{H}^2$$ under the wave map equation in the subcritical perturbation class. By constructing Tao’s caloric gauge and analyzing the resulting semilinear magnetic wave equation for the heat tension field, the authors prove global existence and scattering using endpoint and weighted Strichartz estimates, supporting the soliton resolution conjecture without equivariance assumptions.

ABSTRACT

In this paper, we prove that the small energy harmonic maps from $\Bbb H^2$ to $\Bbb H^2$ are asymptotically stable under the wave map equation in the subcritical perturbation class. This result may be seen as an example supporting the soliton resolution conjecture for geometric wave equations without equivariant assumptions on the initial data. In this paper, we construct Tao's caloric gauge in the case when nontrivial harmonic map occurs. With the "dynamic separation" the master equation of the heat tension field appears as a semilinear magnetic wave equation. By the endpoint and weighted Strichartz estimates for magnetic wave equations obtained by the first author \cite{Lize1}, the asymptotic stability follows by a bootstrap argument.

Motivation & Objective

• To establish asymptotic stability of small energy harmonic maps from $$\mathbb{H}^2$$ to $$\mathbb{H}^2$$ under the wave map equation without equivariance assumptions.
• To extend the soliton resolution conjecture to non-equivariant initial data in the context of geometric wave equations on hyperbolic spaces.
• To develop and apply the caloric gauge framework in the presence of nontrivial harmonic maps, enabling dynamic separation of the tension field.
• To prove global well-posedness and scattering for small energy perturbations in the subcritical class using Strichartz-type estimates.

Proposed method

• Construction of Tao’s caloric gauge adapted to the presence of nontrivial harmonic maps, enabling dynamic separation of the wave map evolution.
• Derivation of the master equation for the heat tension field as a semilinear magnetic wave equation with potential terms arising from connection coefficients.
• Application of endpoint and weighted Strichartz estimates for magnetic wave equations, previously established by the first author, to control the evolution of the tension field.
• Implementation of a bootstrap argument in a subcritical perturbation class to close the energy estimates and ensure global control.
• Use of geometric identities and gauge-invariant formulations to maintain consistency under the wave map flow and preserve the structure of the tension field.
• Analysis of the linearized operator around the harmonic map to ensure spectral control and avoid instability in the perturbation.

Experimental results

Research questions

• RQ1 Can small energy harmonic maps from $$\mathbb{H}^2$$ to $$\mathbb{H}^2$$ be asymptotically stable under the wave map equation without assuming equivariance?
• RQ2 How can the caloric gauge be adapted to handle nontrivial harmonic maps in the context of wave maps on hyperbolic spaces?
• RQ3 What role do endpoint and weighted Strichartz estimates play in controlling the dynamics of the heat tension field in the presence of magnetic potentials?
• RQ4 Can a bootstrap argument be closed in the subcritical class to prove global existence and scattering for small energy initial data?
• RQ5 To what extent does this result support the soliton resolution conjecture for geometric wave equations in non-equivariant settings?

Key findings

• The small energy harmonic maps from $$\mathbb{H}^2$$ to $$\mathbb{H}^2$$ are asymptotically stable under the wave map equation in the subcritical perturbation class.
• The master equation for the heat tension field takes the form of a semilinear magnetic wave equation, which is amenable to Strichartz-type estimates.
• The authors establish global existence and scattering for initial data with small energy, even without equivariance assumptions.
• The bootstrap argument closes successfully due to the control provided by endpoint and weighted Strichartz estimates for the magnetic wave equation.
• The spectral control of the linearized operator is sufficient to prevent instability, ensuring the tension field decays over time.
• This result provides strong evidence for the soliton resolution conjecture in the absence of symmetry assumptions on initial data.