Minimal surfaces of constant curvature in sn

Journal Article (Journal Article)

In this note, we study an overdetermined system of partial differential equations whose solutions determine the minimal surfaces in Sn of constant Gaussian curvature. If the Gaussian curvature is positive, the solution to the global problem was found by [Calabi], while the solution to the local problem was found by [Wallach]. The case of nonpositive Gaussian curvature is more subtle and has remained open. We prove that there are no minimal surfaces in Sn of constant negative Gaussian curvature (even locally). We also find all of the flat minimal surfaces in Sn and give necessary and sufficient conditions that a given two-torus may be immersed minimally, conformally, and flatly into Sn. © 1985 American Mathematical Society.

Full Text

Duke Authors

Cited Authors

  • Bryant, RL

Published Date

  • January 1, 1985

Published In

Volume / Issue

  • 290 / 1

Start / End Page

  • 259 - 271

International Standard Serial Number (ISSN)

  • 0002-9947

Digital Object Identifier (DOI)

  • 10.1090/S0002-9947-1985-0787964-8

Citation Source

  • Scopus