An Inequality Comparing the Dirichlet Energy and the Bienergy of Maps Between Riemannian Manifolds
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866910058669932544 |
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| author | Stepanov, Sergey Tsyganok, Irina |
| author_facet | Stepanov, Sergey Tsyganok, Irina |
| contents | We establish a geometric inequality relating the Dirichlet energy $E_1(f)$ and the bienergy $E_2(f)$ of smooth maps \[ f : (M,g) \to (\overline{M},\overline{g}) \] between Riemannian manifolds. Assume that $(M,g)$ is a compact, connected Riemannian manifold whose Ricci curvature has global minimum $\operatorname{Ric}_{\min}$, and that the target manifold $(\overline{M},\overline{g})$ has non-positive sectional curvature along $f(M)$. We prove that \[ E_2(f) \ge \operatorname{Ric}_{\min}\, E_1(f). \] We further analyze the equality case and obtain rigidity results: equality holds if and only if $f$ is totally geodesic and of constant rank. Applications to maps into Hadamard manifolds are also presented.
To the best of our knowledge, this is the first geometric inequality directly relating the Dirichlet energy and the bienergy of smooth maps. This result establishes a direct connection between the Ricci curvature of the domain and higher-order variational energies. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_15433 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | An Inequality Comparing the Dirichlet Energy and the Bienergy of Maps Between Riemannian Manifolds Stepanov, Sergey Tsyganok, Irina Differential Geometry 53B21 (primary), 53C43 (secondary) We establish a geometric inequality relating the Dirichlet energy $E_1(f)$ and the bienergy $E_2(f)$ of smooth maps \[ f : (M,g) \to (\overline{M},\overline{g}) \] between Riemannian manifolds. Assume that $(M,g)$ is a compact, connected Riemannian manifold whose Ricci curvature has global minimum $\operatorname{Ric}_{\min}$, and that the target manifold $(\overline{M},\overline{g})$ has non-positive sectional curvature along $f(M)$. We prove that \[ E_2(f) \ge \operatorname{Ric}_{\min}\, E_1(f). \] We further analyze the equality case and obtain rigidity results: equality holds if and only if $f$ is totally geodesic and of constant rank. Applications to maps into Hadamard manifolds are also presented. To the best of our knowledge, this is the first geometric inequality directly relating the Dirichlet energy and the bienergy of smooth maps. This result establishes a direct connection between the Ricci curvature of the domain and higher-order variational energies. |
| title | An Inequality Comparing the Dirichlet Energy and the Bienergy of Maps Between Riemannian Manifolds |
| topic | Differential Geometry 53B21 (primary), 53C43 (secondary) |
| url | https://arxiv.org/abs/2602.15433 |