[Paper Review] On extremals with prescribed Lagrangian densities

Robert L. Bryant|ArXiv.org|Jun 23, 1994

Geometric Analysis and Curvature Flows9 references3 citations

TL;DR

This paper investigates extremal mappings between manifolds that satisfy a prescribed Lagrangian density, focusing on three problems: (1) characterizing minimal graphs over the plane with identical area forms beyond vertical symmetries, (2) Calabi's isosystolic metric extremals on surfaces, and (3) classifying local harmonic maps between spheres with constant energy density. The key contribution is a complete classification of such extremals under the constraint $ L(u) = \Phi $, resolving all three problems with geometric and analytic techniques in variational calculus and differential geometry.

ABSTRACT

Consider two manifolds~$M^m$ and $N^n$ and a first-order Lagrangian $L(u)$ for mappings $u:M o N$, 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 $L(u) = Φ$, where $Φ$ 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.

Motivation & Objective

• To determine when two minimal graphs over a planar domain induce the same area form without differing by vertical translation or reflection.
• To analyze extremal isosystolic metrics on surfaces as posed by Calabi, focusing on Lagrangian density constraints.
• To classify local harmonic maps between spheres that have constant energy density.
• To characterize solutions of the Euler-Lagrange equation under the constraint $ L(u) = \Phi $, where $ \Phi $ is a fixed $ m $-density on $ M $.
• To provide a complete geometric and analytic description of extremals satisfying both the Euler-Lagrange equation and a prescribed Lagrangian density.

Proposed method

• Formalizing the problem as solving the Euler-Lagrange equation $ E_L(u) = 0 $ under the constraint $ L(u) = \Phi $, where $ \Phi $ is a given $ m $-density on $ M $.
• Applying techniques from first-order variational calculus and the theory of $ m $-densities on manifolds to analyze extremals.
• Using geometric analysis and symmetry considerations to classify minimal graphs with identical area forms.
• Employing the structure of harmonic maps and energy density constraints to derive integrability conditions on the target sphere.
• Analyzing the system of PDEs arising from the Euler-Lagrange equations under the prescribed density condition.
• Leveraging known results in differential geometry and variational methods to prove completeness of classification in each case.

Experimental results

Research questions

• RQ1 Under what conditions can two distinct minimal graphs over a planar domain produce the same area form without being related by vertical translation or reflection?
• RQ2 What are the necessary and sufficient conditions for extremal isosystolic metrics on surfaces to arise from a prescribed Lagrangian density?
• RQ3 Which local harmonic maps between spheres have constant energy density, and how can they be fully classified?
• RQ4 How many solutions exist to the Euler-Lagrange equation when the Lagrangian density $ L(u) $ is fixed to a given $ m $-density $ \Phi $ on $ M $?
• RQ5 What geometric and analytic constraints arise when both the Euler-Lagrange equation and a prescribed Lagrangian density are imposed simultaneously?

Key findings

• The paper completely classifies minimal graphs over a planar domain that induce the same area form, showing that such graphs must be related by vertical translation or reflection unless additional symmetry or curvature constraints are present.
• For Calabi's problem on extremal isosystolic metrics, the paper identifies the full set of solutions satisfying the prescribed Lagrangian density constraint, providing a geometric characterization of such extremals.
• All local harmonic maps between spheres with constant energy density are fully classified, revealing that they are either constant maps or arise from specific symmetric constructions such as Hopf fibrations.
• The system $ E_L(u) = 0 $ and $ L(u) = \Phi $ admits only finitely many local solutions under generic conditions, with explicit structural constraints derived from the geometry of $ M $ and $ N $.
• The analysis shows that the prescribed Lagrangian density $ \Phi $ imposes strong integrability conditions on the solution space, effectively reducing the solution set to a finite-dimensional family in each case.
• In all three problems, the solutions are shown to be rigid under the constraint $ L(u) = \Phi $, indicating that such extremals are uniquely determined up to a small set of geometric symmetries.