Math @ Duke

Publications [#318271] of Robert Bryant
search www.ams.org.Papers Published
 Bryant, R, 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 2019/08/18)
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: This article 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.


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