On extremals with prescribed Lagrangian densities

TL;DR

This paper investigates the existence and uniqueness of solutions to Euler-Lagrange equations with prescribed Lagrangian densities, focusing on minimal graphs, extremal isosystolic metrics, and harmonic maps with constant energy density.

Contribution

It provides complete solutions to three problems related to solutions of Euler-Lagrange equations with specific Lagrangian density constraints.

Findings

Characterized when minimal graphs induce the same area form

Solved problems related to extremal isosystolic metrics on surfaces

Determined conditions for harmonic maps between spheres with constant energy density

Abstract

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.