Papers Published
Author's Comments:
I prove that the only minimal surfaces of constant
positive Gaussian
curvature in the n-sphere are, up to rigid motion, the
Boruvka spheres,
i.e., the 2-dimensional orbits of an irreducible
representation of SO(3)
into SO(m) for some m less than or equal to n+1. I
rederive Ejiri's classification
of the minimal surfaces of zero Gaussian curvature in
the n-sphere and
prove that there are no minimal surfaces of constant
negative curvature
in the n-sphere. (Partial results had been obtained by
Ejiri.) I also prove
that the only minimal surfaces of constant curvature in
the hyperbolic
n-ball are the totally geodesic surfaces. (That the only
minimal surfaces
of constant curvature in flat space are the planes is due
to Pinl.)
The methods are purely local and depend on analysing the overdetermined system for minimal isometric embedding by organizing the integrability conditions into managable form, so that one can actually differentiate them many times and still have some control over the resulting relations.
The actual results are stronger than these theorems suggest. What I do is classify the harmonic maps with constant energy density from a surface of constant Gauss curvature to the n-sphere. In this form, I have recently generalized these results in On extremals with prescribed Lagrangian densities.
Reprints are available.
Abstract:
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.