Families of proper minimal surfaces

Source abstract

Abstract Assume that $X$ is a connected, open, oriented smooth surface, $B$ is a compact Euclidean neighbourhood retract, and $\mathscr{J}=\{J_{b}\}_{b\in B}$ is a continuous family of complex structures on $X$ of local Hölder class $\mathscr{C}^\alpha$ for some $0<\alpha <1$. We construct a continuous family of $J_{b}$-conformal minimal immersions $u_{b}:X\to \mathbb{R}^{3}$, $b\in B$, properly projecting to $\mathbb{R}^{2}$ and having an arbitrary given family of flux homomorphisms $\text{Flux}_{u_{b}}:H_{1}(X,\mathbb{Z})\to \mathbb{R}^{3}$. In particular, there are continuous families of proper $J_{b}$-holomorphic null immersions $X\to \mathbb{C}^{3}$ and of proper $J_{b}$-holomorphic immersions $X\to \mathbb{C}^{2}$, $b\in B$.