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# Communications in Analysis and Geometry

## Volume 30 (2022)

### Number 9

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

Pages: 1955 – 2009

DOI: https://dx.doi.org/10.4310/CAG.2022.v30.n9.a2

#### Author

#### Abstract

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.

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

Received 4 February 2016

Accepted 11 March 2020

Published 17 August 2023