Localized Curvature Domination and Rigidity of Harmonic Maps
Fuente:
arXiv
Guardado en:
| Autor principal: | |
|---|---|
| Formato: | Preprint |
| Publicado: |
2026
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866912937209233408 |
|---|---|
| author | Stepanov, Sergey |
| author_facet | Stepanov, Sergey |
| contents | We establish a localized Bochner-type rigidity theorem for harmonic maps between Riemannian manifolds. Let $f : (M,g) \to (\overline{M},\overline{g})$ be a harmonic map from a compact manifold. Instead of assuming a global nonpositivity condition on the sectional curvature of the target, we impose a curvature bound localized to the image $f(M)$, expressed via the maximal sectional curvature encountered along the image.
We prove that if the minimal Ricci curvature of the domain dominates this image-dependent curvature bound in a sharp quantitative pinching inequality involving the maximal energy density of $f$, then the map is constant. At the critical threshold, we obtain a homothetic classification: the differential is parallel and the image is totally geodesic.
The result replaces global curvature sign assumptions with an image-dependent curvature domination principle and yields a localized analogue of Yano-Ishihara-type rigidity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_01805 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Localized Curvature Domination and Rigidity of Harmonic Maps Stepanov, Sergey Differential Geometry 53C20, 53C43 We establish a localized Bochner-type rigidity theorem for harmonic maps between Riemannian manifolds. Let $f : (M,g) \to (\overline{M},\overline{g})$ be a harmonic map from a compact manifold. Instead of assuming a global nonpositivity condition on the sectional curvature of the target, we impose a curvature bound localized to the image $f(M)$, expressed via the maximal sectional curvature encountered along the image. We prove that if the minimal Ricci curvature of the domain dominates this image-dependent curvature bound in a sharp quantitative pinching inequality involving the maximal energy density of $f$, then the map is constant. At the critical threshold, we obtain a homothetic classification: the differential is parallel and the image is totally geodesic. The result replaces global curvature sign assumptions with an image-dependent curvature domination principle and yields a localized analogue of Yano-Ishihara-type rigidity. |
| title | Localized Curvature Domination and Rigidity of Harmonic Maps |
| topic | Differential Geometry 53C20, 53C43 |
| url | https://arxiv.org/abs/2603.01805 |