Math @ Duke

Publications [#243410] of Robert Bryant
search www.ams.org.Papers Published
 Bryant, RL, Conformal and minimal immersions of compact surfaces into the 4sphere,
Journal of Differential Geometry, vol. 17 no. 3
(January, 1982),
pp. 455473, International Press of Boston [MR84a:53062], [doi]
(last updated on 2019/08/21)
Author's Comments: This is a short paper in which I use the twistor fibration
of complex projective 3space over the 4sphere to
construct, for each compact Riemann surface, a
conformal and minimal immersion of that surface into
the 4sphere.
The idea is that complex projective 3space has a
natural, SO(5)invariant holomorphic contact structure
and, under the twistor fibration, holomorphic
contact curves project conformally to minimal surfaces
in the 4sphere. As a holomorphic contact manifold,
complex projective 3space is birationally
equivalent to the projectivized tangent bundle of the
complex projective plane. Since every compact
Riemann surface occurs as an immersed curve
in the complex projective plane, it's just a matter of
putting it in general
position (so as to avoid the singularities of the
birational transformation)
in order to get every compact Riemann surface as an
embedded contact curve
in complex projective 3space.
Not every minimal surface in the 4sphere arises
this way (although all of the genus 0 ones do), and I
unfortunately coined the term superminimal
to refer to the ones that do. There are several reasons
to abandon this terminology now and I discourage its
use. It would be better if they were to be called
isotropic, in consonance with the usage in the
theory of harmonic maps.
Reprints are available.


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