Let $(M, g)$ be a Riemannian manifold equipped with a calibration $k$-form $\alpha$. In earlier work with Cheng and Madnick (AJM 2021), we studied the analytic properties of a special class of $k$-harmonic maps into $M$ satisfying a first order nonlinear PDE, whose images (away from a critical set) are $\alpha$-calibrated submanifolds of $M$. We call these maps Smith immersions, as they were originally introduced in an unpublished preprint of Aaron Smith. They have nice properties related to conformal geometry, and are higher-dimensional analogues of the $J$-holomorphic map equation. In new joint work (arXiv:2311.14074) with my PhD student Anton Iliashenko, we have obtained analogous results for maps out of $M$. Slightly more precisely, we define a special class of $k$-harmonic maps out of $M$, satisfying a first order nonlinear PDE, whose fibres (away from a critical set) are $\alpha$-calibrated submanifolds of $M$. We call these maps Smith submersions. I will give an introduction to both of these sets of equations, and discuss many future questions.

On homogeneous spaces, solutions to the prescribed Ricci curvature equation coincide with the critical points of the scalar curvature functional subject to a constraint. We provide a complete description of Palais--Smale sequences for this functional. As an application, we obtain new existence results for the prescribed Ricci curvature equation, which enables us to observe previously unseen phenomena. Joint work with Wolfgang Ziller (University of Pennsylvania).