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Publications [#243410] of Robert Bryant
search arxiv.org.Papers Published
- Bryant, RL, Conformal and minimal immersions of compact surfaces into the 4-sphere,
Journal of Differential Geometry, vol. 17 no. 3
(January, 1982),
pp. 455-473, International Press of Boston [MR84a:53062], [doi]
(last updated on 2024/03/28)
Author's Comments: This is a short paper in which I use the twistor fibration
of complex projective 3-space over the 4-sphere to
construct, for each compact Riemann surface, a
conformal and minimal immersion of that surface into
the 4-sphere.
The idea is that complex projective 3-space has a
natural, SO(5)-invariant holomorphic contact structure
and, under the twistor fibration, holomorphic
contact curves project conformally to minimal surfaces
in the 4-sphere. As a holomorphic contact manifold,
complex projective 3-space 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 3-space.
Not every minimal surface in the 4-sphere arises
this way (although all of the genus 0 ones do), and I
unfortunately coined the term super-minimal
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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