Explicit harmonic and wave maps into variable-curvature surfaces

TL;DR

This paper develops a method to explicitly construct harmonic and wave maps between variable-curvature surfaces, reducing complex PDEs to integrable ODEs, thus providing new solutions beyond classical symmetric cases.

Contribution

It introduces a unified approach to explicitly solve harmonic and wave maps into variable-curvature surfaces, extending beyond classical constant-curvature scenarios.

Findings

Explicit solutions for harmonic and wave maps into ellipsoids.

Reduction of PDEs to integrable ODEs under a natural ansatz.

Applicability in both elliptic and hyperbolic regimes.

Abstract

We construct explicit harmonic and wave maps between pseudo-Riemannian surfaces of variable curvature. For a broad class of target metrics, including nonconstant curvature surfaces such as ellipsoids, the harmonic and wave map equations admit a reduction to integrable ordinary differential equations under a natural ansatz. This yields explicit solutions beyond the classical constant-curvature and symmetric-space settings. The method applies uniformly in both elliptic and hyperbolic regimes.