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Publications [#318271] of Robert Bryant
search www.ams.org.Papers Published
 Bryant, RL, On extremals with prescribed Lagrangian densities,
in Manifolds and geometry (Pisa, 1993), Symposia Mathematica, edited by Bartolomeis, P; Tricerri, F; Vesentini, E, vol. 36
(1996),
pp. 86111, Cambridge University Press, ISBN 0521562163 [MR99a:58043], [dgga/9406001]
(last updated on 2018/03/17)
Author's Comments: 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.
Abstract: Consider two manifolds~$M^m$ and $N^n$ and a firstorder 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 EulerLagrange 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 EulerLagrange 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.


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