A special class of $k$-harmonic maps inducing calibrated fibrations
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866917178893139968 |
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| author | Iliashenko, Anton Karigiannis, Spiro |
| author_facet | Iliashenko, Anton Karigiannis, Spiro |
| contents | We consider two special classes of $k$-harmonic maps between Riemannian manifolds which are related to calibrated geometry, satisfying a first order fully nonlinear PDE. The first is a special type of weakly conformal map $u \colon (L^k, g) \to (M^n, h)$ where $k \leq n$ and $α$ is a calibration $k$-form on $M$. Away from the critical set, the image is an $α$-calibrated submanifold of $M$. These were previously studied by Cheng-Karigiannis-Madnick when $α$ was associated to a vector cross product, but we clarify that such a restriction is unnecessary. The second, which is new, is a special type of weakly horizontally conformal map $u \colon (M^n, h) \to (L^k, g)$ where $n \geq k$ and $α$ is a calibration $(n-k)$-form on $M$. Away from the critical set, the fibres $u^{-1} \{ u(x) \}$ are $α$-calibrated submanifolds of $M$.
We also review some previously established analytic results for the first class; we exhibit some explicit noncompact examples of the second class, where $(M, h)$ are the Bryant-Salamon manifolds with exceptional holonomy; we remark on the relevance of this new PDE to the Strominger-Yau-Zaslow conjecture for mirror symmetry in terms of special Lagrangian fibrations and to the $\mathrm{G}_2$ version by Gukov-Yau-Zaslow in terms of coassociative fibrations; and we present several open questions for future study. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2311_14074 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A special class of $k$-harmonic maps inducing calibrated fibrations Iliashenko, Anton Karigiannis, Spiro Differential Geometry 53C38, 53C43, 58E20 We consider two special classes of $k$-harmonic maps between Riemannian manifolds which are related to calibrated geometry, satisfying a first order fully nonlinear PDE. The first is a special type of weakly conformal map $u \colon (L^k, g) \to (M^n, h)$ where $k \leq n$ and $α$ is a calibration $k$-form on $M$. Away from the critical set, the image is an $α$-calibrated submanifold of $M$. These were previously studied by Cheng-Karigiannis-Madnick when $α$ was associated to a vector cross product, but we clarify that such a restriction is unnecessary. The second, which is new, is a special type of weakly horizontally conformal map $u \colon (M^n, h) \to (L^k, g)$ where $n \geq k$ and $α$ is a calibration $(n-k)$-form on $M$. Away from the critical set, the fibres $u^{-1} \{ u(x) \}$ are $α$-calibrated submanifolds of $M$. We also review some previously established analytic results for the first class; we exhibit some explicit noncompact examples of the second class, where $(M, h)$ are the Bryant-Salamon manifolds with exceptional holonomy; we remark on the relevance of this new PDE to the Strominger-Yau-Zaslow conjecture for mirror symmetry in terms of special Lagrangian fibrations and to the $\mathrm{G}_2$ version by Gukov-Yau-Zaslow in terms of coassociative fibrations; and we present several open questions for future study. |
| title | A special class of $k$-harmonic maps inducing calibrated fibrations |
| topic | Differential Geometry 53C38, 53C43, 58E20 |
| url | https://arxiv.org/abs/2311.14074 |