Nonlinear stability of the totally geodesic wave maps in non-isotropic manifolds

TL;DR

This paper proves the nonlinear stability of totally geodesic wave maps into anisotropic Riemannian manifolds by establishing global existence results for small initial data using advanced PDE techniques.

Contribution

It introduces a novel approach to analyze the stability of geodesic wave maps in non-isotropic manifolds, extending previous isotropic results.

Findings

Global existence for small initial data

Geometric stability of totally geodesic wave maps

Generalization of hyperboloidal foliation method

Abstract

In this article we investigate a type of totally geodesic map which has its image being a geodesic in an anisotropic Riemannian manifold. We consider its nonlinear stability among the family of wave maps. We first establish the factorization property and then formulate the stability problem into a PDE system in a specially constructed chart of geodesic normal coordinates. With a generalization of the hyperboloidal foliation, we establish the global existence result associate to small initial data for this PDE system, which leads to the geometric stability.