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Explicit harmonic and wave maps into variable-curvature surfaces

Anestis Fotiadis, Giannis Polychrou

TL;DR

The paper develops a method to construct explicit harmonic and wave maps between surfaces with variable curvature by reducing the PDE system to ODEs via a traveling-wave ansatz. It identifies a broad class of target metrics $h = A(R) dR^2 - δ^2 B(R) dS^2$ for which first-integral reductions and variable separation yield explicit solutions in both elliptic and hyperbolic regimes. The authors demonstrate the approach with concrete examples, including harmonic maps into ellipsoids and Lorentzian wave maps into hyperboloids, as well as a mixed-signature case, extending known constant-curvature integrability to variable-curvature targets. This provides new closed-form solutions useful as test cases and to illuminate geometric mechanisms in harmonic and wave maps.

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.

Explicit harmonic and wave maps into variable-curvature surfaces

TL;DR

The paper develops a method to construct explicit harmonic and wave maps between surfaces with variable curvature by reducing the PDE system to ODEs via a traveling-wave ansatz. It identifies a broad class of target metrics for which first-integral reductions and variable separation yield explicit solutions in both elliptic and hyperbolic regimes. The authors demonstrate the approach with concrete examples, including harmonic maps into ellipsoids and Lorentzian wave maps into hyperboloids, as well as a mixed-signature case, extending known constant-curvature integrability to variable-curvature targets. This provides new closed-form solutions useful as test cases and to illuminate geometric mechanisms in harmonic and wave maps.

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.

Paper Structure

This paper contains 11 sections, 6 theorems, 37 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Isothermal coordinates and metric conventions
  4. Energy density and Euler--Lagrange equations
  5. Reduction to a first integral
  6. Ansatz reducing the system to ODEs
  7. Proof of Theorem 1
  8. Examples of harmonic maps
  9. Harmonic maps into ellipsoids
  10. Lorentzian wave maps into hyperboloids
  11. A mixed-signature example

Key Result

Theorem 1

Let $a,b,\kappa,\lambda \in \mathbb{R}$ be such that $b^2 \neq \epsilon^2a^2$ and set $t = ay - bx$. Consider the map $u = (R,S): (M,g)\rightarrow (N,h)$, where and Assume that $R(t)$ is a smooth solution of the above equation. Then $u$ is a harmonic map.

Theorems & Definitions (10)

  • Theorem 1
  • Lemma 2
  • proof
  • Proposition 3
  • proof
  • proof
  • Proposition 4
  • proof : Sketch of proof
  • Proposition 5
  • Proposition 6