A generating function of a complex Lagrangian cone in $\mathbf{H}^n$

Abstract

We formulate the space of multivalued branched minimal immersions of compact Riemann surfaces of genus γ ≥ 2 into R, and show that it is a complex analytic set. If an irreducible component of the complex analytic set admits a non-degenerate critical point, then we construct a complex Lagrangian cone in H derived from the complex period map, and obtain its applications as follows: The irreducible component can be divided among some open connected components of non-degenerate critical points, and each connected component admits a special pseudo Kähler structure with the signature (p, q). We induce a sharp inequality between q and the Morse index of a minimal surface which are two invariants of the connected component. Furtheremore, we obtain an algorithm to compute the Morse index and the signature.