Chuaqui, M., Hamada, H., Hernandez, R., & Kohr, G. (2014). Pluriharmonic mappings and linearly connected domains in Cn. Isr. J. Math., 200(1), 489–506.
Abstract: In this paper we obtain certain sufficient conditions for the univalence of pluriharmonic mappings defined in the unit ball of Cn . The results are generalizations of conditions of Chuaqui and Hernandez that relate the univalence of planar harmonic mappings with linearly connected domains, and show how such domains can play a role in questions regarding injectivity in higher dimensions. In addition, we extend recent work of Hernandez and Martin on a shear type construction for planar harmonic mappings, by adapting the concept of stable univalence to pluriharmonic mappings of the unit ball into Cn .

Chuaqui, M., & Hernandez, R. (2023). Families of homomorphic mappings in the polydisk. Complex Var. Elliptic. Equ., Early Access.
Abstract: We study classes of locally biholomorphic mappings defined in the polydisk Pn that have bounded Schwarzian operator in the Bergman metric. We establish important properties of specific solutions of the associated system of differential equations, and show a geometric connection between the order of the classes and a covering property. We show for modified and slightly larger classes that the order is Lipschitz continuous with respect to the bound on the Schwarzian, and use this to estimate the order of the original classes.

Chuaqui, M., & Hernandez, R. (2015). AhlforsWeill extensions in several complex variables. J. Reine Angew. Math., 698, 161–179.
Abstract: We derive an AhlforsWeill type extension for a class of holomorphic mappings defined in the ball Bn, generalizing the formula for Nehari mappings in the disk. The class of mappings holomorphic in the ball is defined in terms of the Schwarzian operator. Convexity relative to the Bergman metric plays an essential role, as well as the concept of a weakly linearly convex domain. The extension outside the ball takes values in the projective dual to Cn, that is, in the set of complex hyperplanes.

Chuaqui, M., & Hernandez, R. (2013). The order of a linearly invariant family in Cn. J. Math. Anal. Appl., 398(1), 372–379.
Abstract: We study the (trace) order of the linearly invariant family in the ball Bn defined by parallel to SF parallel to <= alpha, where F : Bn > Cn is locally biholomorphic and SF is the Schwarzian operator. By adapting Pommerenke's approach, we establish a characteristic equation for the extremal mapping that yields an upper bound for the order of the family in terms of alpha and the dimension n. Lower bounds for the order are established in similar terms by means of examples. (C) 2012 Elsevier Inc. All rights reserved.

Chuaqui, M., & Hernandez, R. (2007). Univalent harmonic mappings and linearly connected domains. J. Math. Anal. Appl., 332(2), 1189–1194.
Abstract: We investigate the relationship between the univalence of f and of h in the decomposition f = h + (g) over bar of a serisepreserving harmonic mapping defined in the unit disk D subset of C. Among other results, we determine the holomorphic univalent maps It for which there exists c > 0 such that every harmonic mapping of the form f = h + (g) over bar with vertical bar g'vertical bar < c vertical bar h'vertical bar is univalent. The notion of a linearly connected domain appears in our study in a relevant way. (c) 2006 Elsevier Inc. All rights reserved.

Chuaqui, M., Hernandez, R., & Martin, M. J. (2017). Affine and linear invariant families of harmonic mappings. Math. Ann., 367(34), 1099–1122.
Abstract: We study the order of affine and linear invariant families of planar harmonic mappings in the unit disk. By using the famous shear construction of Clunie and SheilSmall, we construct a function to determine the order of the family of mappings with bounded Schwarzian norm. The result shows that finding the order of the class SH of univalent harmonic mappings can be formulated as a question about Schwarzian norm and, in particular, our result shows consistency between the conjectured order of SH and the Schwarzian norm of the harmonic Koebe function.
