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Minimal surfaces and deformations of holomorphic curves in Kahler-Einstein

Summary: Minimal surfaces and deformations of
holomorphic curves in Kšahler- Einstein
Claudio Arezzo
To the memory of my friend Giorgio Valli
In this note we give explicit examples of stable nonholomorphic
minimal surfaces representing classes of type (1, 1) in Kšahler- Einstein
4-manifolds of negative scalar curvature. We use the implicit function
theorem for the area functional to deform some holomorphic rational
curves as minimal surfaces for nearby Kšahler- Einstein metrics, and
some results in the theory of deformations of Hodge structures to
prove that the generic of these deformations cannot be holomorphic
for the deformed complex structure. We also show that this strategy
cannot work in the Ricci-flat case, by getting a riemannian proof of
the fact that rigid nodal curves in K3 surfaces can be deformed into
a holomorphic curve in any direction which keeps the class of type
(1, 1).
1991 Math. Subject Classification: 58E12, 53A10.
1 Introduction


Source: Arezzo, Claudio - Dipartimento di Matematica, Università degli Studi di Parma


Collections: Mathematics