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Publications [#243394] of Robert Bryant
search www.ams.org.Papers Published
 Bryant, RL, Minimal surfaces of constant curvature in s^{n},
Transactions of the American Mathematical Society, vol. 290 no. 1
(January, 1985),
pp. 259271, American Mathematical Society (AMS) [MR87c:53110], [doi]
(last updated on 2019/08/25)
Author's Comments: I prove that the only minimal surfaces of constant
positive Gaussian
curvature in the nsphere are, up to rigid motion, the
Boruvka spheres,
i.e., the 2dimensional 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 nsphere and
prove that there are no minimal surfaces of constant
negative curvature
in the nsphere. (Partial results had been obtained by
Ejiri.) I also prove
that the only minimal surfaces of constant curvature in
the hyperbolic
nball 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 nsphere. 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 twotorus may be immersed minimally, conformally, and flatly into Sn. © 1985 American Mathematical Society.


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