Explicit harmonic and wave maps into variable-curvature surfaces
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866917539491086336 |
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| author | Fotiadis, Anestis Polychrou, Giannis |
| author_facet | Fotiadis, Anestis Polychrou, Giannis |
| contents | Explicit harmonic and wave maps are typically available only in highly symmetric or constant-curvature settings, where additional symmetry or integrability structures are present. We develop a reduction framework for pseudo-Riemannian surfaces that extends explicit constructions to a geometrically significant class of variable-curvature targets. For target metrics of the form $A(R)\,dR^2 - δ^2 B(R)\,dS^2$, a geometrically adapted travelling-wave ansatz reduces the Euler--Lagrange system to a solvable system of first-order ODEs. The method applies simultaneously to harmonic and wave maps, treating the elliptic and hyperbolic regimes uniformly within a single framework. As concrete applications, we construct explicit harmonic maps into ellipsoids, Lorentzian wave maps into hyperboloids and the Schwarzschild exterior, and a mixed-signature example, all in genuinely variable-curvature geometries where explicit constructions are substantially less accessible. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_18376 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Explicit harmonic and wave maps into variable-curvature surfaces Fotiadis, Anestis Polychrou, Giannis Differential Geometry Analysis of PDEs 58E20, 53C43, 35L70, 53C50 Explicit harmonic and wave maps are typically available only in highly symmetric or constant-curvature settings, where additional symmetry or integrability structures are present. We develop a reduction framework for pseudo-Riemannian surfaces that extends explicit constructions to a geometrically significant class of variable-curvature targets. For target metrics of the form $A(R)\,dR^2 - δ^2 B(R)\,dS^2$, a geometrically adapted travelling-wave ansatz reduces the Euler--Lagrange system to a solvable system of first-order ODEs. The method applies simultaneously to harmonic and wave maps, treating the elliptic and hyperbolic regimes uniformly within a single framework. As concrete applications, we construct explicit harmonic maps into ellipsoids, Lorentzian wave maps into hyperboloids and the Schwarzschild exterior, and a mixed-signature example, all in genuinely variable-curvature geometries where explicit constructions are substantially less accessible. |
| title | Explicit harmonic and wave maps into variable-curvature surfaces |
| topic | Differential Geometry Analysis of PDEs 58E20, 53C43, 35L70, 53C50 |
| url | https://arxiv.org/abs/2512.18376 |