On extremals with prescribed Lagrangian densities

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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.








Robert Bryant

Phillip Griffiths Professor of Mathematics

My research concerns problems in the geometric theory of partial differential equations.  More specifically, I work on conservation laws for PDE, Finsler geometry, projective geometry, and Riemannian geometry, including calibrations and the theory of holonomy.

Much of my work involves or develops techniques for studying systems of partial differential equations that arise in geometric problems.  Because of their built-in invariance properties, these systems often have special features that make them difficult to treat by the standard tools of analysis, and so my approach uses ideas and techniques from the theory of exterior differential systems, a collection of tools for analyzing such PDE systems that treats them in a coordinate-free way, focusing instead on their properties that are invariant under diffeomorphism or other transformations.

I’m particularly interested in geometric structures constrained by natural conditions, such as Riemannian manifolds whose curvature tensor satisfies some identity or that supports some additional geometric structure, such as a parallel differential form or other geometric structures that satisfy some partial integrability conditions and in constructing examples of such geometric structures, such as Finsler metrics with constant flag curvature.

I am also the Director of the Simons Collaboration Special Holonomy in Geometry, Analysis, and Physics, and a considerable focus of my research and that of my students is directed towards problems in this area.

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