Families of proper minimal surfaces

TL;DR

This paper constructs continuous families of proper minimal and holomorphic immersions on a surface with prescribed flux, extending the theory of minimal surfaces to parameterized families.

Contribution

It introduces a method to generate continuous families of proper minimal and holomorphic immersions with arbitrary flux on a surface, for varying complex structures.

Findings

Constructed continuous families of $J_b$-conformal minimal immersions.

Achieved proper projections to $ extbf{R}^2$ with prescribed flux.

Extended results to families of $J_b$-holomorphic null and immersions.

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 H\"older 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 ${\rm Flux}_{u_b}:H_1(X,\mathbb{Z})\to\mathbb{R}^3$. 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 $X\to\mathbb{C}^2$, $b\in B$.