This manuscript studies some examples of the family of problems where a Lagrangian is given for maps from one manifold to another and one is interested in the extremal mappings for which the Lagrangian density takes a prescribed form. The first problem is the study of when two minimal graphs can induce the same area function on the domain without differing by trivial symmetries. The second problem is similar but concerns a different `area Lagrangian' first investigated by Calabi. The third problem classified the harmonic maps between spheres (more generally, manifolds of constant sectional curvature) for which the energy density is a constant multiple of the volume form. In the first and third cases, the complete solution is described. In the second case, some information about the solutions is derived, but the problem is not completely solved.
Consider two manifolds~$M^m$ and $N^n$ and a first-order Lagrangian $L(u)$ for mappings $u:M\to N$, i.e., $L$ is an expression involving $u$ and its first derivatives whose value is an $m$-form (or more generally, an $m$-density) on~$M$. One is usually interested in describing the extrema of the functional $\Cal L(u) = \int_M L(u)$, and these are characterized locally as the solutions of the Euler-Lagrange equation~$E_L(u)=0$ associated to~$L$. In this note I will discuss three problems which can be understood as trying to determine how many solutions exist to the Euler-Lagrange equation which also satisfy $L(u) = \Phi$, where $\Phi$ is a specified $m$-form or $m$-density on~$M$. The first problem, which is solved completely, is to determine when two minimal graphs over a domain in the plane can induce the same area form without merely differing by a vertical translation or reflection. The second problem, described more fully below, arose in Professor Calabi's study of extremal isosystolic metrics on surfaces. The third problem, also solved completely, is to determine the (local) harmonic maps between spheres which have constant energy density.