A special class of $k$-harmonic maps inducing calibrated fibrations

Abstract

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 $u:(L^k,g)\to (M^n,h)$ where $k\le n$ and $\alpha$ is a calibration $k$-form on $M$. Away from the critical set, the image is an $\alpha$-calibrated submanifold of $M$. These were previously studied by Cheng-Karigiannis-Madnick when $\alpha$ 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 $u:(M^n,h)\to (L^k,g)$ where $n\ge k$ and $\alpha$ is a calibration $(n-k)$-form on $M$. Away from the critical set, the fibres $u^{-1}\{u(x)\}$ are $\alpha$-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.

25 pages, no figures. Version 2: Various minor improvements based on referee's report, including strengthening results showing that the two formulations of the Smith immersion/submersion equations, in terms of calibration forms or in terms of cross products, are always equivalent. Final version, to appear in Mathematical Research Letters. Version 3: Corrected one minor typo