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

We formulate the space of multivalued branched minimal immersions of compact Riemann surfaces of genus $\gamma \geq 2$ into $\mathbf{R}^n$, 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 $\mathbf{H}^{n \gamma}$ 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. Furthermore, we obtain an algorithm to compute the Morse index and the signature.

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The author was partially supported by JSPS KAKENHI Grant Number 15K04859.

Received 4 February 2016

Accepted 11 March 2020

Published 17 August 2023