Title:

Holomorphic curves and minimal surfaces in Kähler manifolds

Our work is concerned with the relation between a complex differential geometric property, namely holomorphicity, and a metric one, namely to be conformal and minimal, of immersions (possibly branched) of Riemann surfaces into Kahler manifolds. A well known theorem (Wirtinger's Inequality) states that every holomorphic surface inside a Kahler manifold is area minimizing w.r.t. variations with compact support. Of course, the converse is not true in general. However, there are important situations, as in the resolution of the Frankel Conjecture by Siu and Yau, when it is. A first motivation for our research is to understand to which extent is the converse true. In Chapter 1 we discuss this problem after having briefly recalled the basic notions and background material that will be needed in the sequel. We first tried to prove some general existence result for immersions into riemannian manifolds, which are area minimizing among classes of maps sharing some topological properties. Following the line of the proof of the existence theorem for minimal surfaces incompressible on the fundamental group, due to SacksDhlenbeck and SchoenYau, in Chapter 2 we prove existence of minimal surfaces incompressible on the first homology group. We apply this result to the theory of Abelian Varieties, and we present here a new proof, completely based on riemannian techniques, of a classical result about the Schottky Problem, i.e. the characterization of the jacobian locus inside the space of principally polarized abelian varieties of complex dimension 2 and 3. A crucial step in the proof of this result is the fact, proved by Micallef, that a converse of Wirtinger's Inequality holds for immersions of closed surfaces of genus 2 and 3 into flat T4 or T6, respectively. As for the Schottky problem, also for minimal surface theory, the situation becomes more difficult as the dimension of the target torus increases. In Chapter 3 we give a unified presentation of an unpublished theorem of Micallef (Theorem 3.4.1) with our research work. In particular we give a very explicit way to construct stable minimal immersions of surfaces of genus r ≥ 4 into flat tori of dimension 2r and of genus r ≥ 7 into flat tori of dimension 2( r  1). The existence of such examples represents a major difficulty in the attempt to apply minimal surface theory to the theory of abelian varieties. In his thesis Micallef proved a converse of Wirtinger's Inequality for isometric stable minimal immersions of complete oriented surfaces into R4, with the euclidean metric, provided that the Kahler angle of the immersion omits an open set of [0,π]. In Chapter 4 we show that this result does not depend on the linear structure of R4, but on a riemannian property of its flat metric, namely the fact that it is hyperkähler. We prove in fact the same theorem replacing R4 with any hyperkähler 4manifold. In the same Chapter we give also a description of known results about the relation between the Kähler angle and the Gauss lift (or the Gauss map, in the case of euclidean space) associated to an immersion. In the last Chapter we go back to the study of periodic minimal surfaces. The results we proved in Chapter 2 and 3 pointed out many natural questions about uniqueness and rigidity of periodic minimal surfaces with some topological constraints. In Chapter 5 we describe a framework for the study of this kind of problems that we believe to be very promising in many different situations, and we study in detail this setting for immersions of surfaces of genus r into flat T2r. Our approach makes transparent a deep connection between algebraic properties of an algebraic curve and riemannian properties of the conformal minimal immersions into some flat torus of a fixed closed Riemann surface. Using previous results of Pirola and classical theorems about algebraic curves, such as the Torelli and the Infinitesimal Torelli Theorems, we give fairly complete answers to the problems about uniqueness and rigidity of minimal maps. In particular we see that these minimal immersions do not share the same rigidity properties as holomorphic and harmonic maps, but nevertheless they generically do not come in families. We are convinced that a deeper study of periodic minimal surfaces in fiat tori from the riemmanian point of view could give some new results in the theory of algebraic curves, especially about the structure of the singular locus of the theta divisors. We believe that our approach gives already some new insight on known phenomena.
