My research is devoted to study the relations between the theory of minimal surfaces and the theory of holomorphic curves in Kaehler manifolds. The basic question we are interested in is the following (see the research project for a discussion on this statement):
Is any stable minimal immersion of a Riemann surface into a Kaehler manifold holomorphic with respect to some complex structure compatible with the metric ?
Because of a classical obstruction to the solution of this problem, it is particularly interesting to consider Kaehler manifolds which admit a large family of compatible complex structures, as the hyperkaehler ones. The simplest examples of hyperkaehler manifolds are the Euclidean 4-space and the flat 4-torus, in which the above problem has been studied by Micallef. I refer to the research project for a discussion on his results. I give now a brief list of the directions that I am pursuing.
In the case of a K3 surface with the Calabi-Yau metric I expect to pro the existence of stable minimal non-holomorphic two-spheres, using a result of Atiyah and Hitchin in the case of a non-compact simply-connected hyperkaehler 4-manifold.
In the case of flat tori of real dimension 2n, n>2, I expect to
prove the existence of stable minimal non-holomorphic of Riemann surfaces of genus g when g>n. I have already proved this result in the case when g=n+1. By previous results of Micallef and Arezzo-Micallef this would give a complete answer to the above problem for flat tori.
Linked with this project many natural questions arise. The first one i the existence of area minimizing surfaces in a fixed homology class. In this direction I expect to generalise a theorem of Schoen-Yau about incompressible minimal surfaces (see the research project for a discussion on this problem). This project is very near to completion.
Again related with the existence problem for minimal surfaces, I expec to prove the following result: in any flat torus of real dimension 4 there exists a countable number of distinct minimal immersions of Riemann surfaces of every genus greater than three. This would follow from a characterisation of which Riemann surfaces admit a conformal minimal immersion in R4 with the minimum number of Jacobi vector fields. We refer to the research project for a more detailed discussion about these results and the link between these and previous research.