# On extremals with prescribed Lagrangian densities

Book Section (Chapter)

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.

### Full Text

### Duke Authors

### Cited Authors

- Bryant, R

### Cited Editors

- Bartolomeis, P; Tricerri, F; Vesentini, E

### Published Date

- 1996

### Volume / Issue

- 36 /

### Book Title

- Manifolds and geometry (Pisa, 1993)

### Start / End Page

- 86 - 111

### Published By

### Place of Publication

- Cambridge, UK

### International Standard Book Number 10 (ISBN-10)

- 0-521-56216-3