Abstract
AbstractWe show that the metrisability of an oriented projective surface is equivalent to the existence of pseudo-holomorphic curves. A projective structure $${\mathfrak {p}}$$
p
and a volume form $$\sigma $$
σ
on an oriented surface M equip the total space of a certain disk bundle $$Z\rightarrow M$$
Z
→
M
with a pair $$(J_{{\mathfrak {p}}},{\mathfrak {J}}_{{\mathfrak {p}},\sigma })$$
(
J
p
,
J
p
,
σ
)
of almost complex structures. A conformal structure on M corresponds to a section of $$Z\rightarrow M$$
Z
→
M
and $${\mathfrak {p}}$$
p
is metrisable by the metric g if and only if $$[g] : M \rightarrow Z$$
[
g
]
:
M
→
Z
is a pseudo-holomorphic curve with respect to $$J_{{\mathfrak {p}}}$$
J
p
and $${\mathfrak {J}}_{{\mathfrak {p}},dA_g}$$
J
p
,
d
A
g
.
Funder
Johann Wolfgang Goethe-Universität, Frankfurt am Main
Publisher
Springer Science and Business Media LLC
Cited by
2 articles.
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1. The Transport Oka-Grauert principle for simple surfaces;Journal de l’École polytechnique — Mathématiques;2023-05-04
2. 3-dimensional Levi-Civita metrics with projective vector fields;Journal de Mathématiques Pures et Appliquées;2022-07