The Goffman-Liu area and Plateau’s problem

Abstract:

Goffman and Liu defined a lower semicontinuous area $A(f)$ for linearly continuous maps $f$ from the disk into ${{\mathbf {R}}^n}$, showed that $A(f)$ is the Lebesgue area when $f$ is continuous and thereby extended the notion of area to some discontinuous maps. With the aid of a simple retraction of the punctured disk onto its boundary, a canonical linearly continuous map ${f_\Gamma }$ of the disk is associated with each continuous closed curve $\Gamma$ defined on the boundary of the disk. When $\Gamma$ is a Jordan curve, it is shown that the discontinuous map ${f_\Gamma }$ has the property that $A({f_\Gamma }) = A(\sigma )$ where $\sigma$ is a continuous map of least area spanning $\Gamma$ from the classical Plateau problem. Finally, the corresponding least area problem in the class of linearly continuous maps is shown to be trivial, that is, the least area is zero in the class of linearly continuous maps.